A binary tree is a common data structure used in software development. It is also a frequent topic in technical coding interviews.

We just published a course on the freeCodeCamp.org YouTube channel that will teach you all about binary tree algorithms and prepare you to use them in coding interviews and programming projects.

Alvin Zablan from Structy developed this course. He has created many technical course, including one of the most popular Dynamic Programming courses on the internet.

In this course you will first learn about the theory behind the algorithms and then you will learn how to implement them with code. The algorithms will be taught with images and visualization to help you really understand how they work.

This type of binary tree will not be covered in the course.

Here are the topics covered in this course:

  • What is a Binary Tree?
  • Binary Tree Node Class
  • Depth First Values
  • Breadth First Values
  • Tree Includes
  • Tree Sum
  • Tree Min Value
  • Max Root to Leaf Path Sum

Watch the full course below or on the freeCodeCamp.org YouTube channel (2-hour watch).

Transcript

(autogenerated)

Binary tree is a common data structure used in software development.

It's also a frequent topic and technical coding interviews in this course, Alvin will explain binary tree algorithms and prepare you to use them in both interviews and coding projects.

Hey programmers, Hamilton from Shruthi, elk to our course on binary trees, I want to show you how you can do well on those technical interviews that have binary tree problems.

So what I have in store for this course, like usual, we'll go over both the theory pines and binary tree algorithms, as well as of course be more practical and come up with a code implement those algorithms really a one two punch.

For every section, we're going to be sure to draw out a pictures and visualize and truly understand the algorithm on the whiteboard notes are feeling comfortable with that, we'll go over to the code implementation.

So in the description and in the timestamps, you'll be able to find the corresponding exercise, you can practice every algorithm that we go over.

In terms of the prerequisites for this course, I'm going to assume that you're not new to programming.

And maybe you've dabbled in some previous data structures and algorithms.

And you're also familiar with some recursion.

But that being said, I'm going to assume you know nothing about binary trees, and even just trees in general.

with no further ado, let's jump right in.

So first order of businesses, let's actually understand what the word tree even means.

And like the programming concept, right.

So when we visualize a tree, a tree contains many nodes, typically we draw nodes as circles, right.

And these nodes can also point to other nodes.

So here I have one node.

And I could point to some other node, right, I could point to any number of nodes.

So here are the circles, I'm going to refer to them as nodes, and the lines or hours between them, I'm going to call edges, right.

So this is an example of a tree.

And trees can of course, come in many different shapes and sizes.

So here's quite a large tree.

Let's try to understand a few terms we can use during our technical interviews.

And this is something I highly recommend, right? helps if you speak the language, it does show that you have proficiency dealing with the data structure, we could store values within the nodes of our tree.

For now I'm just going to put some letters.

When it comes to your programs, you can store any type you want, you can store integers, numbers, or even other objects.

So because we have a tree, we like to use familial relationships.

In other words, I think of like a family tree, right? So if I look at node B, let's say I call them the parent.

If b is the parent, I know that children of B are just the D nodes, right? their parent and child is like a relative relationship.

Let's say I kind of changed my frame of reference.

And let's say I looked at a as the parent, well, that scenario, then B and C are the children of a.

for the case of the C node, if I think of CSM parent, it only has one child, its lone child being the F node.

So feel free to use that parent child relationship when describing the relationship among nodes within a tree.

Another terminal is a lot in the context of trees is the word root, right? So a root is going to be a node that has no parent.

So in this tree, A is the root because a has no parent, right? There are no arrows going into the a node.

On the flip side of that, if I look at the nodes in DNF, we call them leaf nodes.

A leaf node is a node that has no children, right, so d and f have no outgoing arrows.

Typically, in a binary tree, we're going to have a single root and we could have many leaves.

One thing I want to be sure to do is make sure you generalize your understanding of a leaf, right? So in this particular example, looks like every leaf is two edges away from the root, counting the number of arrows from the roots, any leaf, it could be the case that my leaves occur at different levels, right? So I removed that F node, which case C is now a leaf, right? Although a C is on like a different level than D and E, it's not the bottommost level, it is the case that it's still a leaf, right? a leaf node is just a node that has zero children.

All right.

So with all of those terms out of the way, let's hone in on the real topic here.

Right, I talked a lot about trees.

But really in this course, I want to go over the basics of binary trees.

Right? So let's look at that binary part to begin with.

That's really a dead giveaway.

We know binary has to do with the number two.

A binary tree is a tree where every node has at most two children.

Right? So currently on the screen, I have a binary tree.

If I gave a another child, let's say a third node over here of f, this would be a tree but not a binary tree.

Right? It would be like a ternary tree because I have at most three children.

So let's say I kind of remove that extra node.

Now it's back to a binary tree.

And it could be the case that a node has less than two children but still be a binary tree, right? So if I gave C, just a child of f, this is still a binary tree.

It's if I look at any node within my tree, that node will have 01 or two children, and no one has more than two children, right? So you want to be sure to remember that a node in a binary tree could have less than two children.

All right.

So I'll be the first criteria for understanding what a binary tree is right? It's probably the easiest criteria and most children per node will also want to be sure to add into our definition Have a robust understanding is to also remember that, at least for us in computer science, we think of our binary trees as having exactly one root, meaning there should be only a single node that has no parent, right? Typically when we draw, it would be like the topmost node.

So right now this is definitely so far a binary tree.

If I had some other node in this drawing, like G, right, g also has no parents, I would no longer consider this a classic binary tree, right? So watch out for that.

Let's go on to our final criteria, we also need one more ingredient.

And that's exactly one path between the root and any node, right.

So looking at my example to the left, this is indeed a binary tree that meets all three criteria.

And let's say I chose the root and any node, obviously, there's only one root, so we're definitely considering the a node, let's say I chose some random node like E, this is really a binary tree, then we're guaranteed to have only a single path that connects A to E, a path is really just a series of connected nodes I can travel through, right, so to get from the root a to the node E, I can go from A to B to E, and that would be one path.

And that's the only way to get from A to E.

That's what I mean by exactly one path.

Let's say I chose another node like F.

There is also exactly one way to get from the root to F, and that's just a CF, that would be the only path.

If I added some additional connections or edges within my tree, I can get a scenario like this, this would no longer be a binary tree.

For one, we can see that B has three children, but also does not have a unique path between the route in any node.

For example, if I chose a and the note of f, one way to get there is by going ACF.

But another way to get there would be a, b, c, f, right? And so do watch out for all three criteria when you're considering a binary tree.

Awesome.

So let's look at some smaller examples.

And this is where I think some students tend to struggle with their understanding of binary trees.

So if I took out some nodes over here, this would definitely still be a binary tree, right? Just looking at our different rules, right? Let's say the root node A only had one child would still be a binary tree, right? Because the binary tree only demands that we have at most two children, we still have one root, and we still have only one path from root to any node.

Let's say I have the smallest tree that has a node, which would be the singleton tree, this is still considered a binary tree, right has one route.

And there's not really any pass to be had here except the route to itself.

And every node does have at most two children.

Here we have zero children, right? The final edge case we want to think about is what happens when we have no nodes, right? We consider this as the empty tree.

This is a very special case.

And we should consider any empty tree as actually being a binary tree.

That's going to be very useful when we come up with some algorithms later on, right.

So a common edge case off to consider is what happens when we have an empty tree, that's a tree with zero nodes, right? Let's look at another example.

Let's say I had this structure to take a moment look at it, and see if this meets our criteria.

This would not be considered a binary tree.

So we look at your different criteria.

Looking at the first one, it does have at most two children per node, so that's okay.

But we can't identify exactly one root here.

Remember that a root node is a node that has no parent, right? If you look at every node in this picture, every node has exactly one ingoing arrow.

So that means every node has at least one parent, right? So it doesn't meet that criteria.

And furthermore, there is not exactly one path between the root and any node, because we have a cycle within the structure.

So for example, let's say I was starting at a right and I wanted to travel to see, one way to get there would be to go A, B, C, that'd be one path.

But I could also have another path or I just go around twice and go A, B, C, again, there would actually be an infinite number of paths in this scenario.

Because of that, this is definitely not a binary tree.

And there kind of many reasons for that.

If I had a different example, looking at this one, now I need criteria one and two, right, I have at most two children for every node.

But I also still have exactly one root node here I would consider z the root node.

However, there is not one path between the roots in any node and because I have that cycle.

So these rules are really worth remembering, they're really going to help you solve a much more difficult binary tree problems.

And if you remember these three rules, you can identify different problems in a binary tree framework.

In other words, the hardest problems that you'll encounter in your interviews are problems where they don't tell you straight up what data structure you're dealing with, you're just gonna have to notice the pattern.

In other words, what if I gave you some structure like this, take a moment and figure out if this is a binary tree.

If you look closely, this does meet all of the three criteria to be a binary tree.

I can just treat a as the root because it has no parent.

I can treat nodes D and F as the leaves because they have no children.

I just drew this in a pretty interesting way.

Right? If you really range things in a classical binary tree sense would look like this.

More importantly, this has the same relationships as a previous drawing, I just show it in a top down fashion, right? So no matter what do you understand these three rules for a binary tree, right? Because sometimes may not be as explicit as a nice triangular shaped drawing.

Awesome.

So we've talked a lot about the theory of how to, you know, look at and reason through some binary tree definitions.

Let's go ahead and start to talk about how we could represent binary tree programmatically.

In other words, how can we do this in some code? Well, no matter your programming language of choice, I think you're going to represent these as objects.

In other words, every node here is going to be some objects.

So it could be like an instance of a class, the properties only to store within this object would be the current value.

So I need something to store like the A of my current node.

But I also need to refer to some children.

So we'll also need some left and right pointers to the children, right, those are just going to be some properties on that object.

In a classic binary tree is very common to refer to the two children using a left and right direction.

Notice that some nodes here, like the C node, they're only going to have one child, right? c only has a node dot, right? It does not have a left.

So we're gonna have to use some empty value, like null or undefined.

To represent a child that doesn't exist, right? Especially for a node like he, he has no left and also no, right.

So here's what we'll do, let's go ahead and hop into my text editor.

And I'll show you how you could represent a binary tree programmatically.

So transition, here we are in my text editor.

Let's start by creating our node class, right.

So you'll be able to follow along any language that you choose, I'm going to do mine in JavaScript, right.

So I think the best way to present a node is just to use some class.

So hopefully familiar with some classic class syntax, let's create a node class, there's going to be a quick constructor, I think would be valuable to take in the initial value that's going to be stored within the notes, I'll take the ns a constructor argument, I can just set this dot Val to that value, and also need two more properties, one for my left, so I can say this dot left, I'm gonna initialize it to null, that means by default, a node will have no left child the same way.

By default, a node will have no right child, right? So when you use no to represent children that don't exist over here, awesome.

That's all you need to create, you know, a baseline binary tree, right, we're going to use this class a lot during the course to test our algorithms.

So I'm going to create a node and eventually wire up a tree.

Well, I'll just call my constructor few times, I'll create different nodes, I'll just store them to some variable names.

And of course, a new node to create new instance of node, and I'll store some value inside, I'm going to store some characters inside, I'll create a bunch of these different nodes, what we'll want to be sure to do is make sure that you also set their pointers properly.

So I'll give each of these some different values ABC, D, E, also do F.

And then I'll just manually for now set their pointers properly.

So what if I did syntax like this is since I'm in JavaScript, I can easily assign these pointers, I can say, A's left is going to be B.

I can say A's right? is C can also say B's left D.

and then B's right is E.

And maybe Finally I can say C's right is F.

So by doing these assignments, I'm connecting some nodes together, I should end up with a structure like this right? Have a as a root, because nothing points to a it has no parent.

But a has two children of the NC.

And then along with that, B has two children of D and E.

Finally, C just has a right child of f, this is actually the same tree that we looked at a lot during the whiteboard session, right? And that's how you can represent that same tree programmatically, right? So I always think it's valuable to also try to visualize your trees, right? Obviously, we created this tree in a very manual way, probably in the long run will create applications to just maintain and create trees dynamically during some input.

But for now, we'll start by creating all of our trees in this very kind of static way.

Right? And so with that, now that we have an understanding of how to represent a binary tree, let's go head back to the whiteboard.

And I can show us our very first algorithm.

Hey, programmers, Alvin here, right? Now I want to go over this depth first values problem.

So this will be a good review of the depth first traversal algorithm.

So what do you want to do in this problem? Well, what we want to do is take in a binary tree, and in particular, your function is going to take in the root of the binary tree.

And just recall that given the root node of a binary tree, we know that that node is going to have pointers to its left and right children, which may point to other nodes.

However, if let's say a node does not have left or right child, then its point is going to be set to No.

So that's how I represent our binary tree programmatically.

But for now, let's just stick to the visual represent of our tree.

So for a depth first traversal, we could start with the root node of a, what we'll do is we'll just add that to some collection.

And we're going to have to maintain these in a very particular order.

And then from there, according to a depth first traversal, I could go to B.

And here's where we make our really important decision, right, we can either go to C, or D, if I'm doing a depth first traversal, I need to go deeper in the tree before I move laterally.

So that means I go from B to D.

And once I bought them out at D, there's nowhere deeper, I can go from D.

And so now I move laterally to the E node.

And this pattern continues, right? I don't have anything deeper from E.

So now I go to C.

And then from C, I go to f.

So this would be a depth first traversal on this binary tree.

Notice that it goes A, B, D, E, C, F, and again, the really important characteristic is we must go deeper in the tree until we can't anymore and then we can go across the tree.

That being said, How can we actually managed to implement this algorithm.

If you're familiar with depth first, traditionally, then we know that it's going to use a data structure like a stack.

So let's get situated over here.

So we'll trace through that again.

But this time, taking a really close look at how we can use a stack to accomplish that direct order.

And so just recall that a stack is a sequential data structure where we can only add things to the top of the stack and remove things from the top of the stack.

So a really important characteristic is we can't really insert or remove elements from anywhere but the top of the stack.

So when I add things to the stack, I was added to the top like this.

When I remove things, it's also from the top like this.

Alright, so when I begin at this depth first traversal algorithm, I'm going to start with the root node of a.

And by default, I really just take that root node and I just store it on my stack.

So it's the only thing on my stack right now, I'll use these rectangles represent my different stack frames.

And really in my program, that would just mean storing the actual instance of node or some pointer to it.

Cool.

And then from there, I actually begin my main algorithm.

So the main flow of a depth first traversal, we'll check if the stack is empty.

Right now the stack is not empty, because I have at least one element.

And so what I do is I start by removing or popping the top element of my stack.

So I'm gonna remove the A, and I'll label that as my current node being explored.

And when something leaves the stack, then I can consider it being visited right now.

So I need to list out my values, because that's the whole point of this problem, right? So by now I've just visited the a node.

At this point, I can look at that nodes children now.

So I look at the a node in the tree, and I see that it has a B child on its left and a C child on its right.

And from there, I push or add those two children to my stack.

So I'm going to put c first, and I put B afterwards.

And notice if I push my right child first, followed by my left child, that makes it such that my left child is at the top of my stack, which means that I would hit them next.

And that actually ends on my first iteration of this depth first traversal.

And now I asked that same question is my stack empty, it's not.

So I removed the top of my stack, I'll call it B, my current, which means I could print it out or insert it into my values list.

Nice.

From there, I consider B's children, B has two children.

So I add them both, I push E, followed by pushing D.

That ends that iteration.

Now something interesting happens, my stack is still not empty.

So I know I removed the top, so I call D, my current, I add D to my list of values.

But if I look at these children, it actually has no children, right, and so there's nothing to add to the stack from here.

And so technically, I finished this iteration.

And now I still have stuff on my stack.

So now I pop he off the top of my stack.

Same thing as before, right, I add to my values, he has no children.

So I'm done with that iteration.

Finally FC, I removed C from the stack, and I printed out in my values.

And then at this point, I see that the C node only has one child, so I'm just going to push the children that exists so I would push the F node onto my stack.

And that's a really important thing to remember when it comes to implementing your depth first traversal you're going to have to check if your children exist before you add them to your stack.

And now on the final final iteration, we have the F node we remove it from the stack, we add it to our list of values f has no children.

And so I finish this iteration and now my stack is empty and I will just exit right once the stack is empty.

I know that I must have traveled through the entire binary tree.

So there we have it and we do get the correct output over here.

So just by using a stack and just obeying the rules of the stack, that is we got pushed to the top and removed from the top we will get the correct order.

important thing to remember is you should add your values into your like values list whenever something leaves the stack.

That being said, What can we say about the time and space complexity of this algorithm? Well, it's actually pretty straightforward.

Let's say we define n as the number of nodes in this binary tree, then we can say that the time complexity of this is O of n.

And why do I say that? Well, we're just going to add that every node eventually to our stack.

And that nodes also going to leave the stack exactly once.

So it's not like we're double visiting any of the nodes.

So I'm guaranteed to just run in O of n steps.

In a similar way, we can see that the space complexity is O of n, the only thing we stored was really the stack, which is a linear data structure.

And we know that we're not going to put ever more than n things on the stack.

So overall, we have a linear time and space solution for this depth first traversal problem.

All right, I think we have a good understanding about the approach for a depth first traversal.

Now let's go ahead to the walkthrough video.

And we'll implement this one in a few ways together.

Hey, programmers, Alvin here, right.

And I want to go over a JavaScript solution for this depth first values problem.

And so we're going to do here is really just implement that same strategy that we traced through in the approach video, pretty much to the tee.

And we'll start with the iterative version.

And that means of course, we will solve this one in two different ways, right, we're gonna solve it iteratively first, and afterwards, we'll go ahead and solve it recursively.

Alright, so let's jump right in over here.

Like we said, in the approach video, we know that the iterative version really relies on us creating a stack.

And for most of your programming languages, you can just go ahead and use your like array data structure for that.

So I'm going to use just a plain old JavaScript array, I'll go ahead and name this one stack.

And we can just use an array to represent a stack as long as we stick to two particular methods, right, I'm going to use array push, which adds to the end of an array, and also array pop, which removes from the end of the array as well.

So I'm going to consider the end of my array to represent the top of my stack.

Cool, we'll go ahead and do is initialize the stack with the root node on top of it.

Nice, then I can have my main loop for the algorithm.

So we know that we need to keep running the algorithm while there's stuff on our stack.

So I can just check while stack dot length is greater than zero.

So while I have at least one element on my stack, there is some work to be done here.

And that will step through a single iteration of this algorithm, we know that we start interation by just removing the top of our stack, in other words, stack dot pop in JavaScript, that will also return us the element that we just removed.

So I'll call that my current const current.

And that is going to be an instance of nodes, it's going to be one of these objects.

Cool.

So now that I remove this element from the top of my stack, let's say for now, I just I don't know, print it out, maybe we'll kind of debug this one as we go.

So console dot log, current dot Val, because I know that every instance of node has a dot Val prop on the inside.

And then from there, I need to add this nodes children.

So you might go ahead and guess that to do that, we could just do stacked up push, I'm going to push the left child, one out here, so node or current dot left, followed by the right child.

But we also need to make sure that these children exist, right? Look at the prompt.

There could be instances, for example, like this C node over here, this C node only has a right child, but has no left child, right.

So what I don't want to do is push C's left, because that'd be pushing no onto my stack, which would give me an error later on.

Right? So I want to only push the children if they exist.

And I need to individually check if the left exists and push it and also if the right exists and push it, I'm just going to add some guard statements for both of these.

So we'll go ahead and insert will say, all right, if the current has a left and right side, current dot left, called justice for the right hand side, cool, maybe we can inline this, if you prefer.

Alright, so I'm going to only push the children if they exist.

And that should actually be the heart of our depth first algorithm.

So we're not putting our values inside of an array like the problem asks, but we should at least be able to see the correct order of printing here, because I'm going to print a nodes value as soon as it is removed from my stack.

So I'll go ahead and actually maybe bring in maybe one literal test case, so I could just steal this little stub over here.

And this will just print out the values.

Right, so I'll test this I can hit Run, if I hit Run, it's just going to execute this file just like a script.

So it's not going to run any automatic test cases, just going to run my file as is.

So if you want to test it in this way, kind of in a manual way, you're gonna have to be sure to call your function.

I do have a console dot log on the inside.

So I'll just run Run this manually.

And when I do that should see some output here looks like node is not defined because I forgot to bring in my class definition.

So I'll do that.

give that a go.

Alright, so looking at our output, we didn't get our exact depth first traversal, like we expected, right? I got ACF, B, D, if you take a look at what we really printed out, we technically did print out a depth first print, but we favored the right hand side.

So we did a, c, f, and then B, E, D.

So let's say we really wanted the left to right version of this, and all you have to do is flip the order that you push the children, right, it should be clear by looking at these two lines, right? When I add my children looks like I push the left child, and then the right child.

If you push the left child first, and then you push the right child afterwards, that means that the right child's going to be on top for the next iteration.

So if you do it like this, this will favor the right hand side and travel through it first.

But if you did it like this, now you'd be actually iterating to the left branch first, toward the right.

So with that small change, let's just see what our print is.

And there we have a nice AB de ECF.

However, in this problem, they want us to actually return those values in an array.

So I can kind of get rid of this little Manual Test over here, I don't need this class definition anymore.

And now instead, I'll just gather up all of my values in some results array.

So I'll say, maybe result over here starts as an empty array.

And then as I pop elements from the top of my stack, instead of logging them, I'll go ahead and do result dot push, and I'll push that value into the result array.

Once I'm done with this while loop, that means my entire tree has been explored.

which case I can return, make sure I spell it right, return my results.

Cool.

So this should be a nice solution, it's actually run the test cases on this.

give that a go.

So it looks like we're doing pretty good.

But for this very last example, looks like we're getting can't read property Val of na, and that's running test case 04.

So if you actually go into the prompt, those test cases are actually laid out explicitly over here.

So I want to go to zero for now let's look at the zero for test case.

They go ahead and they pass my function, no, right kind of representing like an empty tree or an empty node.

And we can kind of trace it what happens here for our code.

Let's say that root was null.

That means when we initialize our stack, our stack literally contains like a null value.

So it's not even an instance of note just now.

So that means when I entered this while we're going to check, you know, Do I have anything in my stack, and I do because my stack length is one.

And then when I pop, the top element of my stack current is going to be no, then on this line 14 it says null dot Val.

And that's actually where our code explodes, right on line 14, right? You can't reference property Val, of null.

And so to handle this scenario, we want to make sure that we actually never allow anything No, to enter the stack.

It's kind of why we had this guard over here.

mentioned also be true for even the top level route, in case they give us the empty tree.

kind of seems like a corny, you know edge case.

But this is a pretty common one, when you actually go into your interviews, right? What happens if you have an empty input.

And so I'll just guard that explicitly, I'll go ahead and check, hey, if my route, so if my entire tree is empty.

So if route equals null, then what I'll do is just return an empty array, right? Because that means there are no values inside of it.

And that actually is the expected answer, according to this, right should return an empty array.

So let's give it a go now should be pretty good.

Nice.

And there, we have our nice iterative solution for this depth first values problem.

Alright, so here's what we'll do.

That was one solution.

Let me show you another way.

let's implement this.

And that would be our recursive flavor.

So I'll just kind of redefine it down below.

And I actually recommend practicing both versions, because they're going to serve as like the basis for which you solve a lot of different tree problems.

And so when I think about the recursive version, like all of our recursive code, I must think about the simplest case of my input, and that will act as my base case, right? So in the case of a binary tree, the simplest tree you can be given is going to be the empty tree or just a no root, right? So it's not even going to be about a node that's single node right inside of a binary tree.

It's going to be about an empty tree, right, a tree with zero nodes.

I'm going to check if my route is no, and I have an empty tree.

And I think about this base case, as if it's its own input, because it really is right and so if someone They asked me to give them back in array of all the values in the empty tree, it's still the case that this must be an empty array, right? Because there are no values, there's even any nodes and the empty tree.

And then from there, I can generate my actual recursive call, right.

And I know what I call my stuff recursively.

That means I have to reference a function and again, go ahead and invoke it.

And I'm going to call upon root dot left, as well as root dot, right? So this would, this call over here would give me back an array containing all of the values in the left subtree.

And this will give me an array of all the values in the right subtree.

And so let's say I see these into their own variables respectively, you don't have to do this part.

But I'm kind of a fan of it, especially if it's the first time implementing this.

So I'll call this my left values.

I'll call this one my right values.

And here's how I'm able to kind of quickly create recursive codes all about taking when I call is a recursive leap of faith here.

So let's say we're stepping through test 00.

And so this is our tree visual, right? So we know that our root is going to be a node.

So when I actually make this top level call, this base case does not fire.

And so I make my recursive calls, right? I know when I do depth first value, or values, rather, let me fix that depth first values of root dot left, that means I'm going to be passing in the B node, right, and here's how I have to actually pretend my code behaves, I'm just going to get back the correct result from this call, right? So if I passed in the B node into this call, what I expect back is the full array representing the depth first traversal of that subtree, starting at B, right.

So if I got correct data over here, what it would look like is just B, D, and E.

That would actually be the full depth first traversal, from this subtree, right.

And I'm kind of doing that just mentally, or I know the expectation for this algorithm, it's going to be a similar story for my right call, right? If I'm at a So A is still my route, and I passed in my right child See, to this call, what I expect back is just an array of CF, which would be the depth first traversal of that left subtree.

And now that I have, you know, my left subtree values and my right subtree values, I have to think about how to combine it all together, right? How do I work myself into that output by combining both of the results from my children, right, what I need to do is really just take myself put it in the array, followed by my left children, followed by my right children.

So I can do some nice JavaScript syntax for that whenever returning the right when I throw myself in there, so I'm going to throw in a root dot Val, that's myself.

So that'd be like the a node that I just take all the values in my left result, put them in here and just use the spread operator to unpack that array.

And I'll do the same thing for the right values of CF, right.

So dot dot, dot, spread out right values just so I return a single array, and this pattern itself does match this output, right, kind of taking the leap of faith and assuming correctness from these recursive calls over here, right, I have a, and I plug in substitute B, D, E, and then for the right, I have CF.

And even before we run this, I'm not going to, you know, assume too much JavaScript knowledge.

So maybe you're kind of new to this spread operator.

So in general, super quick aside over here, let's say had an array of some stuff.

So I'll say, peeps there and some people over here.

So we'll throw in fleabay, we'll throw in Jason, throw in Raj, throw in heavy.

And what I can do is whenever I have an array, I can just use a spread operator to kind of unpack that array.

So for example, I can say const will say new peeps.

And I'll create a new literal array.

So whenever I say square brackets, it gives me a new array literal.

And what I'll do is, I don't know, put an element at the front, we'll call it Alvin.

That'll spread out the elements of peeps, right? So just imagine I took all of the elements here and just remove the brackets.

That way I don't add any initial nesting.

And then at the end, I don't know I can add someone else.

Like, right? Well, console dot log, what new peeps looks like so I'm not gonna run the test cases, we're just gonna run this file as a script, just so we can review this spreads index.

So if I give that a go, we kind of make this bigger.

Notice that I have correctly spread things out right? Have Alvin at the front, and I have all of the things from peeps in the middle, right, followed by Brian at the end.

So you can do this.

You can also use like the concat method on arrays.

You're probably going to see me use this spread syntax a lot because I'm kind of a fan, right.

But with that in mind, let's go ahead and actually test this code.

So we'll run all the test cases.

Nice.

And this is another working solution for our depth first values.

And it should be pretty natural that we can solve this problem also recursively.

So I know that the iterative solution for this depth first traversal requires a stack data structure.

And I know whenever I write recursive code, under the hood, your programming language is actually going to track all those recursive calls, using the call stack, right? That call stack behavior gives you the same type of ordering, which is really convenient.

Cool.

All right.

So before I send you off on the next problem, I want you to actually practice both solutions for this depth first values code, it's going to be very necessary to actually master some later concepts that are coming right up.

So do spend some time on this practice makes perfect, right, and I'll catch you in the next one.

Hey, programmers, Alan here, right? Now I want to go over this breadth first values problem.

So it's really just going to be a little variation off of the depth first one we just did, of course, but it's really important that moving forward for all of our tree algorithms and other data structures, that we have both versions down pat.

So in this problem, we're going to take in a binary tree, once again, very classic binary tree structure, this time, I want to return an array or a list of all the values according to a breadth first traversal order.

So a breadth first traversal starts with the root node of a, so that's nothing new.

And then from there, I could go to B.

So right now I have my a node, and also my B node.

And here is where we diverged from our previous problem in a breadth first traversal.

And breadth just refers to like the width of something I travel across before I go deeper.

So in our breadth first traversal, I'm gonna move to C and not D for now.

So I add my C node c value to my list.

And then from here, once I finished this entire level, and there's nowhere to go across the tree anymore, now I could go downward to the next level.

Now I have D, and then E, and then F.

So they're really important distinction here is the breadth first traversal starts with ABC, whereas the depth first traversal would have started A, B, D.

That's a really, really important distinction.

And so how can we go about implementing the breadth first version of this? Well recall that the depth first used a stack data structure, well is the case that the breadth first is now going to use the queue data structure, really just the partnering version of that structure.

And so let's kind of step through that in a more programmatic way.

So I'm going to use and track my queue.

And just recall that a queue has no sense of direction to have the back of my queue and the front of my queue, things enter the back of the queue, and they leave the front of the queue.

Right.

And so no one gets a skip line either.

So this gives me a nice fair ordering, think of a queue as just like waiting in line at checkout at a grocery store or something.

So how do I begin my algorithm, I'm going to start by initializing my queue with the root node.

So I'm just gonna start with a on my queue.

And then from here, now I can begin my main algorithm.

So my main algorithm should check on every iteration is my queue empty, right now it's not because I have at least one element.

And so I remove the front element of my queue.

And so we know that a would be removed, and we label it as our current.

A node being explored when something leaves the queue and is marked as our current will say that now it's being visited.

So I would add a to the running list of my values.

Then from here, I need to look at his children.

He has two children B and C, I need to go ahead and add them into my queue.

And so let's say I added B into the queue first, followed by C.

Notice that C must enter behind B, right.

And now I finished this iteration.

At this point, I have the next iteration of my algorithm, I take the front element of my queue, and it leaves, right So b is now my current, I add B to my running list, then I look at B's children of D, I push D followed by he behind it.

At this point, I have another iteration to do right.

And it's so on so forth.

From here, see leaves the front of my queue, I add it to my list of values, and I look at sees children see only has one child.

So for that child that exists, I go ahead and add them to the back of my queue.

And it's really important that I added to the back right at this point had my next iteration.

At this point I continue, my queue is still not empty, so I still have some stuff to remove.

So D leaves in front of my queue, I add it to my list of values.

Since D has no children, there's nothing to add here.

next iteration, he leaves the front of my queue, I added to my list of values, no children, so I don't need to add anything.

And finally f leaves the front of my queue, I just add it to my list of values and again, f has no children.

So by now my queue is empty, which must mean that, hey, I must have completed the algorithm, right? There are no more nodes to explored, I explored and added every single node to my list of values.

All right, and this output actually looks correct, right, I get abcdef, just like we say.

But what about the complexity of this algorithm.

So let's take a closer look over here.

Well, we know from the get go, that n is going to be the number of nodes in this binary tree.

So that will be the term for our sides of input.

And we can say that the time complexity is simply just O of n, roughly, because we know when it comes to visiting these nodes, and you know, using our loop, we're going to add every node to the queue once, which also means that that node is going to leave the queue also wants, so it's not like we're double adding a node to the queue, right, we're not going to double visit any of these.

In a similar way, the space complexity is going to be O of n at most, because, you know, we're just going to add at most to all of the nodes into our queue.

And in general, it's probably going to be less than Oh, event space in terms of how much space we use in our queue.

An important thing I'll just mention right now that in regards to the time complexity of this one, here, we're going to say that the time complexity is O of n.

And that's if we can assume that adding something now to the queue runs in constant time.

And removing something from the queue occurs in constant time.

So depending on how you implement this, if you use actually, a built in an efficient queue data structure, you will get this O of n time complexity for our breadth first.

But if you use a less optimal, like data structure, maybe not a perfect queue, then you might have an actual worse complexity.

So again, this time complexity of O of n assumes that we have a maximally efficient queue that has o of one add and remove operations.

With that being said, I feel pretty good about coding this one up.

I'll catch you in the walkthrough video, and I'll show you a few different ways that's implement this one.

Hey, programmers, Alvin here, right now I want to walk through the JavaScript solution for this breadth first values function.

So hopefully, you just finished the depth first version, in which case, this one should be pretty much a cakewalk.

But of course, we will still go through the motions of it, because it's going to be useful to solve much harder problems later on, as you always hear me say, Alright, so to tackle this one, we'll start with our queue data structure, right? So nothing fancy in JavaScript, what I would do is just use an array and just stick to using very specific methods, right? So I'm going to create an array, I'll call it my cube.

And we're going to initialize that queue with the route on it by default.

And we'll also guard against upfront is imagine they gave us an empty tree to begin with.

So in other words, the initial route is no, which case it would be like this test 04.

And they just want us to return an empty array.

So I'm going to guard for that explicitly.

So I'll check Hey, is my route No.

And if it is, just go ahead and return an empty array like that.

Nice.

Then from here, I need to start in my main loop from algorithm.

So I iterate a while my queue is not empty, just like we spoke about in the approach video, right? So while q dot length is greater than zero, then keep on going.

Now I begin a single iteration of this algorithm by just removing the front of my queue, right? So it's really up to you which methods you use, you just need to make sure you remove from one end and add to the opposite end.

So for me, I'm going to treat index zero as if it's the front of my array, I'm going to treat the last index as if it's the back of my array, right? So if I want to remove the front and say, array dot shift, or for me right now, Q dot shift, that removes the front also returns to me, that element, so I'll call that my current note, x.

And then from here, I need to add this node children into my queue.

And that'll give me my main flow for this breadth first traversal.

Right.

So I'm going to go ahead and check, hey, if my left child exists, so if, let's say current dot left, that is not know, then what I should do is go ahead and push that into my queue.

So I'll say q dot push my current dot left, and it's just going to be symmetric for the right hand side, of course.

So if the right exists, then push the right into the queue.

Just like that.

Awesome.

And so that looks pretty good.

And then from there, I actually want to store the nodes I visit in an array for the final return just like the problem asks, so nothing too fancy.

I really want to insert some code over here about and so I'll create a result array, I'll call it values, sort of empty whenever something leaves the queue.

I'll just take that and push it into my values.

So I say values dot push.

I'll push that current Val into my queue.

And do note that, what you can't do is just take like the queue and treat it like your final return value, it must be a totally separate thing, right, I use the queue just for the sake of traveling through in a breadth first order.

And the order that you visit is actually derived from the order in which things lead the queue.

So that's why I'm doing it over here, right, as soon as something leaves a queue, that's what I considered visited.

So I added to my values list.

Nice.

And after my while it's done running, I'll go ahead and return my entire values array.

So let's give this a test.

I'm gonna run all the test cases, what we get, have a few test cases to pass.

And there we have it.

Here's a nice iterative solution for our breadth first traversal.

Just something I want to mention, by the way, let's look at the prompt.

So looking at, I don't know, like the very first example, in this kind of rendition of breadth first, what they asked us to do is really give us a breadth first traversal, that moves from left to right, so notice it goes A, B, C in terms of the resulting output, and not a CB, right, those would both be technically correct, start to a breath first.

And then you can kind of choose depending on what problem you're solving, whether you want to go left to right or right to left.

For me, because I push the left followed by the right, that means the left is going to leave the queue first, right, because remember, the queue is just like a line and checkout.

So if someone enters first, that is the left enters first, they going to be served before the right go.

And you can just flip these around.

And I'll give you the right to left universal goal.

So depending on what your problem warrants, you can always manipulate that code a little bit.

And this is actually probably like the only solution you can have.

You know, maybe aside from just some superficial changes to like the code style, this is going to be your go to and only go to solution for a breadth first traversal on a binary tree, right? A common mistake I see people try to do a lot is there is not really a straightforward way to implement a breadth first traversal recursively.

And that should make some sense, because a breadth first traversal needs a queue order right needs to use a queue.

If you write any recursive code, you know, under the hood, it's using a stack.

And so that stack versus queue is really just going to fight against you.

And you're going to have a really tough time trying to implement the correct ordering.

So always just writes the iterative version for your breadth first traversal.

Alright, so I recommend before you hop into the next video, make sure you're able to write this code on your own, because we are going to level up difficulty a little bit, but I'll catch you in the next one.

Hey programmers, Alvin here, right now I want to walk through the approach for this tree includes problem.

So the premise of this problem is I'm going to give you a binary tree, and also a target value to look for.

I just want you to tell me True or false? Is that target value found within the binary tree? So for this particular example input, the answer is obviously true or Yes, right, you could definitely find e within this binary tree for give you another target value like j, you would respond with false, right? Because that value j is nowhere to be found within the binary tree.

And really what I'm asking you to do here is we're stepping through the canonical like breadth first search and depth first search problems.

And I think I'll walk through both approaches for you right now.

So let's say I was testing or I wanted to trace through rather of the input where target was IE, how can I go about attacking this one, right? We know that in most of these problems, when they give you a binary tree as input, they're only going to give you really the root node.

But that being said, if you have access to the root node, then you know you have access to all nodes that comprise of the tree, right.

So what I can do is just perform any of my traversal algorithms.

So maybe to get going.

Now let's just do either the breadth first search or the depth first search iterative style.

So I think for this trace, I'll stick to the breadth first version, which means we're going to use a queue, right? So as we trace through this, I'm thinking about this one iteratively.

Right now, hopefully you recall from our previous problems when it comes to your breadth first traversal, you start with your root node on your queue.

And when something leaves the queue, you mark it as your current right.

And here's where I work in the new logic for this, this particular problem.

When something leaves the front of my queue, I'm going to check Hey, is that current value the same as my target value, so is a the same as E.

It's not.

So I have not found the thing I'm looking for, right? And what I can do is now consider his children.

Right? So I look at his children.

They both exist, I add them to the queue.

So I'd be to my queue that I add see to my queue.

And now I'm on to my next iteration right? The front end element B leaves a queue, and I have to check, hey, is B, my target, it's not right my target t.

So I keep running, I look at B's children.

So I take that D and added it, I take the E and also add it.

And this process just continues, see leaves front of my queue is see my target, it's not.

So keep going, I would add C's children to my queue.

So I just add F to my queue.

And this process continues.

And so d leaves the front of my queue I check is d my target, it's not, and D has no children to add, right? Recall that in our breadth first traversal, I only add the children if they exist, because I don't want to add any, like null pointers into my queue.

And so I just continue my algorithm, right, I still have stuff in my queue to check.

And finally, when he leaves the front of my queue, I check Is he my target.

And indeed it is right at this point, I've just confirmed that he is found inside of my binary tree.

So what I can do is just end my algorithm by like returning the true value, right? There's no point of actually looking through the rest of the tree, because I already figured out that, hey, my target value is indeed within the tree.

Right? One thing you might notice is, it looks like I only checked for IE once it left the front of my queue, and not when it was initially added to my queue.

Technically, you could have returned early when you added it to the queue.

But we'll kind of see when I walked through the full code for this one, it would end up with cleaner code if you just checked for your target value when something leaves your queue.

So we'll see that when we go through the code.

It's just a small implementation detail.

That being said, for this iterative breadth first strategy, what can we say about the complexity? Well, if we define n as the number of nodes, then we know that the time complexity is just O of n, it's really just a classic breadth first traversal.

So nodes are going to enter the queue once and leave the queue once that's open, right.

And again, that's considering if we use a efficient queue, right, where our queue add and remove operations run an O of one constant time.

So we have linear time.

And for the space complexity, it's also going to be O of n linear, right? Because we're just going to store our nodes within the queue.

And so we just looked at the approach for a nice iterative breadth first solution, not to this problem, but I want to show you the depth first version, and the depth first version, that would be recursive, right? So obviously, you could write the iterative debt per solution, which case you just use basically the same code we just spoke about, but just use a stack instead of the queue.

But let's try to solve this one.

recursively, right.

And the reason I think it's really important to expose yourself to recursion like this is, as we move to more complex topics, you're going to find this style of recursion.

Very, very useful, right? So we have the same input, right? Let's say My target is E, and I have the same binary tree.

And now I want to check recursively is E within this tree? And so how do we start attacking this? Well, we're going to need to think about our base cases, right? So we're gonna have really two types of base cases, we'll have like the affirmative base case, meaning Hey, we found a match.

And so whenever I counter a node, whose value is he, or whose value matches my target, then I'm gonna have that node or that recursive call, return true, right? So that's my, like, affirmative base case.

So I'm just gonna plug that return value in visually in my tree, you know that this II nodes gonna return true.

And now I'll think about my negatory base case, right? Four times where I call upon an empty node, or the null node, I should return false.

And recall from our previous lectures, we said that we're going to sometimes represent explicitly our null nodes.

So I'm going to fill those in, for example, the C node in my picture would have a left child that is null.

So I'll kind of draw that one explicitly.

And I know that a node like that is going to return false.

Because logically, I shouldn't be able to think about all of my recursive calls as if they're their own subproblems.

Right? So if someone asked me, Hey, can you find this E value within an empty tree? Right? a null node represents an empty tree? The answer to be no, I can't find the value in an empty tree, because there's nothing there.

And so from there, I'm also going to draw explicitly all of those similar null nodes, null pointers, right? So it looks something like this.

Notice that from the E node, I don't need to give it a null left and right, because I already said that that node is going to return true, right? So that's why I draw my tree like this.

And I have all of those false return values.

Cool.

So we just labeled all of the base cases, right as the leaves of our tree, and recall that a leaf is just a node with with no children, right? And from here, how do I actually combine all of these Boolean values to get the true at the very top, I know for my top level color, that is at the a node, or the a recursive call, I need to get back that value of true, right? And the logic we should use is really just to the logical OR so how does this one work.

So let's start evaluating our return values and combining them at the parent.

So this should be somewhat of a similar pattern to like the tree sum problem I showed you.

And so what you could do is focus on this D node over here, this node has values ready on its left and right has two falses.

And when it actually gets those falses returned to it, it just should do the logical OR so it's doing false or false, which evaluates to false, right? Which means that hey, in this subtree rooted in D, I cannot find the value, which makes sense.

That's why it's false.

I'll continue this process.

If I look at this B node, now this B node has values ready on its left and right.

And they're going to bubble up a little bit.

And I take the order of them so I do false or true, which evaluates to true.

And this process continues everywhere in the tree, right? So at this f node, false or false, is me false.

At the C node, false or false gives me false.

Finally, at the top level root of A, I do true or false, give me back a final true, which is indeed the correct answer.

So hopefully, you realize how similar the solution is to our previous tree some problem right in the tree, some problem, I combined my left and right, child return values by doing the arithmetic addition.

But this time around, I just need to use the Boolean operation of or right, so by just adjusting the type to Boolean, we have a very, very elegant solution that shout out George Boole.

And so with that, I think let's go ahead and implement this one in some code, and I'll show you it using the interest of flavor or using maybe a breath for us, and also the recursive version, which is my personal favorite, using this recursive tree structure.

Hey, programmers, welcome back.

Alright, and I want to go over a JavaScript solution for this tree includes problem, right? Well solve this both depth first, and also breadth first.

And I think we'll start with the iterative breadth first version.

And so this is really just going to be implementation of the classic breadth first search algorithm, I'm just gonna lay down my classic a breadth first traversal code, and then just add some conditional logic afterwards, right? So you should be familiar with it by now.

But for my breadth first traversal, I'm going to use a cube.

So I'll say const, q equals an empty array, are really an array with the route thrown on the inside.

And from there, I have my main algorithm, right, so I loop while my queue is not empty.

So while queue length is bigger than zero, and keep on going.

And on a single iteration of this traversal, what I do is Q dot shift, so remove from the front of my queue, the front of my array, remove the first element.

And I'll save that into a variable called current.

So that will be just an instance of node.

And for now all why don't I just build up my solution slowly, I'll just, I don't know, console dot log, what current dot Val is.

But once I consider this node, what I want to do is really add its children into my queue only if they exist, though, right? So in general, I'm going to write like q dot push, I'm going to add to the back of my queue, so I get a nice cue order, push the left child, so current left, likewise, the right child.

But imagine that I have an asymmetric node or just a leaf node, right? Something like C would only have a right child, so it's left would be no.

So if current is see, then I'd be pushing No, into my queue as the left, which is no good, right.

And so I need to guard here and only push the children if they exist.

So something like this, hey, if the current has that, corresponding left and right, then I can go ahead and push it.

So this is looking pretty good.

And that should actually be the main traversal part of the code.

So I'll just test this manually.

So very manually, so maybe just this call, so I'm not gonna return Booleans yet.

We'll build up to that.

But I should at least see my values printed.

If I kind of run this manual test case, I'll bring in my node class, I should see the values printed in a correct breadth first orders, that means A, B, C, D, E, F, recall that breadth first traversal travels across our tree before going lower, so I must finish a level before traveling to the next level.

So let's just run this manually as a script.

See what we get here.

A, B, C, D, E, F, nice, so I'm getting the right order of traversal.

And now I can work in I think, the conditional logic because they want us to return Booleans over here.

So pretty straightforward stuff.

We'll go ahead and check once something leaves the queue.

I can check If the current nodes value is equal to my target, then I've found the thing I'm looking for.

So just return true, right, you're done, you don't need to travel through the rest of the tree because you can return true.

But on the flip side, if my value that I'm currently at is not the target value, that I must continue looking through the rest of the tree, right.

So if I finished the entire while loop, that means I've traveled through the entire tree.

And I never found the thing I'm looking for, I should return false.

So I need a late return false over here are really common mistake people tend to make is, what you don't want to do is just do like, else return false here.

So this is going to be wrong.

Right? Because this would only check like the very first notice normally check the root, and then check if the root is not equal to the value.

If it's not, you just return false, which is not really useful, because it could be somewhere else in the tree.

So you're gonna need that late return false pattern over here.

But with that, I think we're going to pass some of the test cases, as well run the test cases by hitting that test button.

And there's probably one scenario we did not for C.

Cool, and there it is, right? Can I read property value? No, we're failing tests.

055, look at that spec.

Test five gives us a null node as a root, right? So just the empty tree, you cannot find the B character inside of the empty tree.

So return looks like false over here.

And so I can just handle that one explicitly.

I'll go ahead and check at the top.

If my route is no, right, if that's the case, then just return false.

I can't possibly find any target in an empty tree.

Right? The reason our code was failing before is if root is null, then I initialize my queue with no.

And when that thing is pop, and that null is popped, or rather shifted from the queue in the first iteration, then I check null dot Val, which is an illegal JavaScript operation.

So I'll run all these test cases now.

Nice.

And there, we have our breadth first solution for this tree includes problem.

All right, now let's work on the recursive depth first version of this, it's actually my personal favorite, because it utilizes a pattern that I think is quite elegant, and one I use a lot for much more difficult problems.

So like we said, in the approach video, if you haven't watched the approach, you definitely want to check it out.

For this recursive version, right? what I should do is check Hey, if my root is null, if I have the empty tree, basically, then just return false, right, because I can't possibly find my target and empty tree, right, that's just a given.

And from there, I know I'm going to have the general shape of some just depth first traversal code, which means you call the same function because it's recursive.

That's what recursive means.

And you pass along your route dot left and a separate call your route, right.

And when I should be sure to do is Don't forget to pass in your target, write the target that you're looking for, it never changes.

And I know that these two these two calls are going to give me back boolean data.

And I know that the Boolean I get back from like this call would represent whether or not I found the target and that subtree, right.

So this gives me the result of if I found it in the left subtree.

Or the result if I found it in the right subtree.

And I can just do the logical OR on both of these, right.

So if I find it in either subtree then return true.

So I could just write in line return true over here, right? So let's say it's in my right subtree, then this left hand side invites to false.

And this is true, in which case, this entire thing evaluates to true.

And let's say in a bad scenario, let's say it's not found in either subtree.

So this evaluates to false.

this right hand side also evaluates to false, false or false gives me false.

However, one thing I need to be sure to add is also an additional base case, after I check if my route is no, what I want to do is then also check, hey, maybe this route I'm currently at, maybe it actually has the target, right? So if route value is equal to the target, then you're also done, except you can return true.

Cool, and that was very reminiscent of our approach video.

So let's give this a shot.

You should be able to pass nice, and I love how clean this code is.

I will admit, you know, it's pretty tricky.

If you're not a fan of writing recursion, in which case, I'll totally convinced you into being a fan of recursion.

But notice how kind of clean this code is really leverages recursion.

One very important thing I want to bring up is it's really keen that I put this base case on line 26 after the null check, right? So let's say I flipped the order of these.

Let's say you did this.

I believe that would not work out always because it assumes that you even have a route Alright, so I'm feeling a test 00.

But in general, if you look at this code, let's say that route was no, right? You're gonna start by checking null dot Val, and I'll throw an error, right? Can I read property valve? No.

So you want to actually always lead with your base case that checks if your root is null, right? Because that should guard really the entirety of our code.

So this is actually the code you want.

Over here.

Alright, programmers.

So that's all I got for this tree includes solution, I want you to practice both versions, and I'll catch you in the next one.

Hey, programmers, Alvin here, right? Now I want to go over the approach for this tree sum problem.

And so in this problem, I want to take in a binary tree, just like we've been doing as of late, this time, the values within the nodes of this binary tree are going to be numbers, right, what I want you to do is compute the total sum of all the values in this tree.

So for this example, we should end up with 25.

And so hopefully, you're kind of gathering how to attack this problem, especially given you know, the algorithms I've been showing you as of late.

So if you've been studying our problems, and are there hopefully, you know, a straightforward solution to solve this one, we could of course, just use any type of traversal algorithm.

So we can use either a breadth first or a depth first traversal.

And we can just add all these values into a running sum.

And of course, we probably initialize that sum to zero, right? So the iterative breadth first or depth first solution, I think, is very straightforward, right.

And I'm not going to really step through that approach with you over here, I think you have that one down pat, especially if you've done the previous two problems.

But what I will show you here is actually how to solve this one recursively, which would be a type of depth first traversal, because we know that hey, depth first traversal relies on a stack.

And if you do something recursive, it is utilizing the underlying call stack.

So I will get a similar type of ordering.

And so when we trace through this type of solution, this recursive solution for this tree, some problem we're going to do is try to be very explicit, I think this is really helpful, especially if you struggle with recursion, right? So given this binary tree, I know that for particular nodes of my tree, like the four node, it does not have a left child, right.

And I know like programmatically, what that means is like that nodes dot left pointer points to like, no, or it's a null pointer.

And so I'm just going to draw that explicitly, sort of like this.

Again, they'll just help us really understand how our recursive code performs on this type of input.

And so if I know that I can put like an unknown load to the left of four, then it could be the case that other nodes like the leaves have to in one, they also have both left and right children that are also know.

So if I drew all of my null pointers explicitly, it would look something like this.

Cool.

And this is really something that helped me really get comfortable with recursive problems, especially those on binary trees.

And what we're doing now is, we know that when it comes to solving any problem recursively, it's about writing a base case, that is like the simplest version of our input.

And here, I'm going to argue that our base case needs to be about the null node, right? The null node basically represents no node at all, or represents the empty tree in a sense.

In other words, if someone asked me to calculate the sum of a null node or the sum of an empty tree, then to me that some would just be zero, to write elegant recursive code to try to think about your base case as a problem in itself, right? So if someone gave me an empty tree, that is a null node, I would return the total sum of zero.

And what that means is I know that all of these null nodes, they would return zero as their computed sums.

Nice.

And then from there, how can I use this information to actually build up my main solution? Right? So let's go ahead and target this for node to the left.

And what I have to do is figure out Hey, know what is this subtrees total sun, and I can compute that given the values to the left and right.

In other words, if I return those two null values returns to zero.

Now I just have to do zero plus four plus zero, which just equates to four, right? And that is actually correct in itself, right? So what green number above a node represents the total sum of that subtree.

And so it is a case that, hey, the total sum of that four node is just four.

And I would do something similar for this node over here, right? When these base cases return to their color or return to their parent, I just have to add my left and right child together along with myself.

So zero plus two plus zero, gives me to not do that same thing.

For this note 11 We're here.

And now things get interesting, right? It's a calculate the total sum rooted at this 11 subtree, I would just return these values to the top.

And now I do four plus 11 plus two, which gives me an answer of 17.

And if you do a quick spot check, right, this 17 does represent the total sum, just in this subtree, right, and we'll carry on over at this one node, we know that this is going to return just a value of one.

And now at this four node, I can compute its total sum.

And recall that this four node had a left child of zero, right? Because it has a null node.

And so when I compute the sum, I get five, which is perfect.

And finally, at the main route, now I bring up these two values.

And I compute 17 plus three plus five.

And that gives me a final answer of 25.

Which is indeed the correct answer.

Right? So that's how we should really think about our recursive algorithms for our binary trees, right? I kind of draw it out, I think about the base case, which is typically not always but typically about the null node.

And then from there, I figured out how a parents can compute its result given its children's answers.

Alright, so if we take a look at the complexity of this, it's pretty straightforward.

We'll go ahead and define n as the number of nodes in our input tree, in which case, I can see that my time complexity is just O of n, right? We know that we're going to make a recursive call just for every node of the tree.

And we're not going to call duplicate nodes, right, we're going to have just one call for every single node.

And within any single node, we're just going to do some simple arithmetic, right, we just had roughly three numbers together.

So it's not like we're going to have like a loop inside of our calls at all.

In a similar way, we'll say the space complexity is O of n, just because we have that implicit, a call stack space, because we are going to solve this one recursively.

And so we're solving this one in O of n time and space, which would be actually a maximum efficient a solution for this problem.

And so with that, I think, let's head over to the walkthrough video, and I'll show you how to code this one up.

Hey, programmers, Alvin here, right? Now I want to walk through the JavaScript solution for this tree some problem, I think this time around, and I'll jump right into the recursive version, right? So we'll start recursively.

And we'll begin with our base case, right? My base case is always about the simplest version of the input, where I just know the answer without needing any additional calculations, right? I know that the simplest tree here that I could be given is going to be the empty tree, right? I have the empty tree.

That means my root is no.

And if your root is no, you have the empty tree.

What is the sum of the empty tree? Well, kind of inherently, it's just zero.

So I'll be my base case.

Nice.

And then from there, I'll think about how I can compute my answer, given my children results.

So I need to find the results of the some of my left subtree.

And also the some of my right subtree.

So just call recursively.

So tree, some of root left, as well as tree some of root.

Right? Cool.

And I know that these two calls, they return numbers, right, representing the sum of my left subtree.

And the sum of my right subtree.

As the parent that is so rude, how can I find my total sum? Well, it just is myself.

So root out value, plus everything in my left subtree plus everything in my right subtree.

Right.

And we'll go ahead and just test this out the gate.

But this is nothing fancy, really elegant code to look at the test cases here, we do have some a diverse output.

So this code does work on trees of all different shapes and sizes.

Something that helps me you know, really believe the magic of recursion is to just analyze the assumptions here, right? So let's say that I'm stepping through tree some and my root is this three over here, right? So I checked the base case, is this three node? No, it's not.

So I must make the recursive call, right.

And I know that when I break down this code recursively, I know my route that Val is going to be like three, so I'm kind of corresponding, this comment with the thing below, right? And then when I make the recursive call on tree, some root dot left, that means I'm asking for the total sum of this subtree starting at 11.

So I'm looking at the 11 for negative two subtree.

Right, if I take the total sum of just that left subtree, it looks like it's going to be just 13.

It's now I'm saying plus 13 over here, right? If I do the same thing on the right, subtree I know that this call is for this four node, right all of its children, that should just magically return by the power of recursion that should return five, right? I'm going to assume that that recursive call just works.

So you're out the five, right? If I take the total sum of these, three plus 13 1616 plus five is 21.

That would give me the correct answer, right.

So to have some confidence in my recursive code, I just write the code.

And then I assume correctness from my recursive calls.

And I figure out how I can take those sub valleys for my children, combine it with my own value, and that should be my final answer.

So here is the recursive version of this solution, which would be some sort of a depth first traversal, technically, because it's recursive.

And if you'd like, I can also show you the iterative version.

So let's just do that.

And I always practice usually my problems doing both the recursive as well as iterative version, if it's reasonable, and is really reasonable here.

Hopefully, you're familiar with our breadth first code by now, right? So I'm just gonna start by checking, hey, if you're given some root that is already know that that's kind of an edge case, I'll just go ahead and return zero, right? Because as some of that empty subtree, or that empty tree, rather, is just zero, that can begin my main code.

So while my queue is not empty, so I'll start with a Q, who begins with the Route Route, I want to loop while my queue is not empty.

So while q dot length is bigger than zero, I'll begin a single iteration of my while loop by just shifting out the front of my queue.

So in JavaScript, for me, that's q dot shift that removes the front element, and also returns it to me.

So I'll save in current.

And then from there, now that I have this current, I know that contains the value that I actually care about.

So what I would do is go ahead and track some running sum.

So I'll say let some member say let total sum equals zero.

And as I remove something from the front of the queue, I'll take it to value and add it to my total, something that will accumulate all of the values right into a sum.

And from there, I just need to add my classic logic of checking for my children, right? If they exist.

So if I have a left, so if current dot left is not equal to know, then what I'll do is add that left child into my queue.

So q dot, push my current dot left, again, make sure you're following your cue rules, right, so shifts removed from the front, push adds to the back, it's really important that my shift and my push or my add and remove methods work on opposite sides of my queue.

Otherwise, it's not a cue, right? So that looks good to go.

I'll write something symmetric.

For the right hand side over here.

By the end of my function, I'll just go ahead and return my total sum.

Nice, this would be my go to enter diversion.

course, it's technically a breadth first, and we'll give that a test.

Nice.

And here we have that solution.

So one thing to note, I think it's pretty important to be consistent how you implement these algorithms, I swear possible.

So you'll notice that I always am a fan of a writing my like processing logic for a node.

Here I consider like the processing logic, where I add the value of my current node into my total sum.

I do that processing logic, typically when a node leaves the queue, and not when the node is added to the queue, and just ends up being less repetitive that way, for example, I've seen some people write code like this, where it's like, Alright, well, you know, I guess since I'm adding my left child into my queue, that means it exists.

So not only will I push it, but I'll add it so I'll do I don't know total sum, plus equals current dot left dot Val, right, so I'm kind of adding the child's value as soon as it's added to the queue.

And you can write something symmetric for the right hand side, you can see that because of this, I kind of have to double up all of my code, which does not make it super clean.

And I think something that is even worse than that is well now you actually never added your root value into the your, your total sum.

So you'd have to begin your total sum as root dot value at this point, I'm like, I kind of just hate this code, right? So you can run this, this would probably work.

So this code totally works, but it's not what I think is the best design for this pattern.

So I'd rather write it like this, right? consume your values when they leave your queue because you know, eventually, everything's gonna leave, right? So I'll prefer it at least like this.

Cool.

So before you go on to the next problem, recommend you practice both versions.

I know of this problem.

So do it recursively do it iteratively have them both down pat because we're going to increase the difficulty right now.

Hey, programmers, Alan here right now.

One To go over the approach for this tree min value problem.

So just like we've been doing as of late, we're going to have to take in a binary tree as input into our function.

And we want to do is figure out what the smallest value within the tree is.

That is we want to find the minimum value.

So we can totally assume that the tree is going to be non empty, right? And so given that information, looking at this particular tree example, it's pretty plain for us to see that the smallest number in the tree is just this three.

Alright, so our function just needs to return of the number three.

So it's plain to see that the answer is three.

But how can we come up with an algorithm to do this for us? Well, we should look to our tools in our binary tree toolbox.

And that would mainly just be either a breadth first or a depth first traversal.

One obvious way to solve this one is to use either a depth first or breadth first iterative piece of code that does a traversal and travels through the tree, then you just need to maintain some outer variable to track the current minimum.

And whenever you hit a node that is smaller than your current minimum, then you replace that minimum variable.

And by the end of the code, you should have the true minimum.

So the iterative version, I think, is pretty straightforward.

And I'll be sure to walk through that when we get to the code walkthrough.

But for now, in terms of our approach, visual, I think we'll step through this one recursively, right, because it ends up being pretty succinct code.

And so let's take a look at this tree.

And let's pretend that we filled in all of the null pointers, like usual, right? So these dark nodes over here represent nodes that don't exist, meaning that there are no, right.

So for example, if I look at the three node, its left child is no, but its right child is 12.

And that's going to be very useful for us to at least visualize because I should make a base case about those null nodes, right? So I know in the long run, what I want to do is figure out what the smallest value within the tree is.

And so a great default value for our null nodes would be to return infinity.

And here's the reasoning why I know that I should not really consider any of the actual null nodes.

And so if I have them return positive infinity, I know that positive infinity is definitely not going to be the minimum compared to any numbers that are actually within the tree.

So we'll start to do is just fill in all of these infinite values.

In other words, every time we call upon a null node, or a null pointer, we're going to return infinity.

So I'd fill all of these in.

Now using those infinities as our basis, we can actually construct our return values.

So looking at this for node to the left, what I have to do is consider both of its children that it received, right, and also the value within the current node.

And so what I do is I check, alright, compared between infinity for an infinity, what's the smallest value among them? Well, I know positive infinity is a very large number.

So really, the smallest number here should be four.

And so this call should return four, right? This node should return for similar story for this 15 node.

Once I have my left and right return values ready, I do a comparison with those values against myself, right? So I check, what's the minimum among infinity? 15 and infinity, the minimum is 15.

I think it's gonna interesting now that we're at evaluating this 11 node, right? Now I have some actual numbers.

And so I do this comparison, right? I compare my left subtree value, which is four and my right subtree value, which is 15.

And also myself, which is 11.

So among 411, and 15.

What's the smallest number? Well, that would just be the four, right? So this node should return four.

And if I do a quick spot check, this makes some sense, because I have this answer for above the 11 node, that's telling me that, within that subtree rooted at 11, the smallest value was four, which is totally correct.

Now we'll just go ahead and valuate the rest of this tree.

Looking at this 12 over here, I know I can't evaluate the three because I need a value on the right hand side ready.

So I must evaluate this 12.

And I just do the comparison, right? What's the minimum among infinity 12 and infinity? Well, let's just definitely 12.

Finally, at this three node, I take the minimum of infinity three and 12, the minimum there is three.

And finally at the ultimate root, I do the comparison.

And I check what's the smallest among four, or five and three? And the answer is going to be the three, right, which is exactly what we expected.

And so just by adding a little variation to our classic depth first recursion, we can totally solve this problem.

The logic we need to use to combine our child sub solutions into our main solution is to just do a comparison, right? So overall, at every point of this tree, we asked ourselves, right, given the smallest value in the left subtree and given the smallest value in the right subtree.

I compare those to myself, and return the smallest among those three.

And then when it comes to the complexity of this, it's pretty straightforward.

We'll go ahead and say that n is the number of nodes within our input tree.

And we know that we're going to have a call for every node of this tree.

So that seems to be an O of n time.

In terms of the number of recursive calls, right, and we know within any particular recursive call, it sounds like we're just going to do some conditional logic, right, just find the minimum among three things.

Since three is a constant number, I'm pretty sure that we're going to have just an overall time complexity of O of n, considering the number of calls that we make.

Now, that being said, we'll also have O of n space complexity adjust due to the call stack that we use for baseline depth first traversal.

And that seems to be an optimal solution for this problem.

Because what we'll do, we'll go ahead and code this one up, and I'll catch in the next one.

Hey, programmers, Alvin here, right now I want to go over a JavaScript solution for this tree min value problem.

And of course, we'll solve this one a few different ways.

I think I'll start by showing y'all maybe some iterative solutions.

And so we'll just do some classic depth first and also breadth first, along the way, right.

So you know, you're going to need to iterate or hit every node within your binary tree.

So just go ahead and set up either a stack or a queue.

So I'll choose to start with a stack over here.

So I'll say Kant's, stack equals empty.

And one thing I'll have to be aware of is looking at the assumptions in the problem, they tell us that we can assume that the input tree is not empty.

And so that means I won't need to add a leading if statement checking if the top level root is null.

So I can just go ahead and initialize my stack with the root node inside.

Cool.

And then from there, you need your main loop to iterate through the different nodes.

So I'm going to loop while my stack is not empty.

So all stack dot length is bigger than zero.

Cool.

So if that's the case, and there are still some nodes to visit, and begin a single iteration of this algorithm, let's say depth first, by doing stack dot pop, so I removed the top of my stack, go ahead and call that my current.

And then from there, of course, we need to check if our children exist.

And if they do, I'll add them to the top of my stack, right? So I'll write a pattern like this, I'll say if, let's say current dot left, that is not equal to No, then I can go ahead and push that existing child on the stack, right? So stack dot push, and I'll push current left retinas a few times by now should it be second nature, I'll do some same code for the right hand side.

And this is kind of my baseline of a depth first traversal, right.

But as I look at this current node and its value, I want to by the end of my loop, have access to the minimum value within the tree.

And so because I need to find the minimum value in this function, we're going to use a variable that I'll update over time, right to track the current smallest thing I've seen.

So I'll initialize this as let smallest, I'm gonna have to think about a good default value for this.

And so I'll go ahead and initialize this to actually positive infinity.

So if you're unfamiliar with some JavaScript, infinity is basically just a number that is guaranteed to be bigger than all other numbers, except for itself infinity, right? And the reason I'm choosing a big number, like positive infinity, for my default value for the smallest variable, is because when I see any actual values of my tree, I know they're guaranteed to be less than infinity.

So these numbers would replace my infinity, just gives me a good initial value, right? That's a pretty common pattern, right? If you're looking for the minimum thing, and typically your default value is positive infinity.

If you're looking for a maximal thing, then your default value might be zero or negative infinity, depending on the problem you're trying to solve.

And so with that default value, where do I actually want to replace it, right.

And I'll choose to do that right after or something leaves my stack.

That way, you only have to say it once.

If I wrote it, when I added my children into the stack, what I'm saying twice, right, which is kind of annoying.

So I'll just check, hey, if current value, so be sure to compare the actual number value, current value is less than the smallest, then I've just found a new smallest, right? So just replace that variable with current value.

Of course, by the end of our while loop, we should just return that variable, right? And we should have the true minimum.

So let's give that a shot, run these test cases.

And this is our rendition of an iterative depth first, right? Awesome.

So if you wanted to switch it up, you know, writing the breadth first is super trivial in terms of the change we make, you can just make sure you do stack dot shift, right, because that would remove the front element of your queue now and if you push then you're still adding it to the back.

And I guess you should probably rename this guy to queue.

So very minute difference.

Well, tests that just give that a go.

Cool, passes all of them.

Nice.

And one thing to note.

So Alex in JavaScript, there is no immediately when I recorded this video, there is an immediately available, like optimal queue data structure just baked in, like you can't import it through a standard library.

And so when you actually do the array dot shift method, that line itself is going to run an O of n time, right, because if you remove the front element of an array in JavaScript, it will have to actually shift all the other elements over one position.

In other words, index one becomes index zero, index two becomes index one, and so on.

And so technically, if you implemented your breadth first exactly like this, just using a regular array, and you shift out, then this would probably be technically like an O n squared solution, which isn't the fastest, but we'll be able to pass the specs, typically n squared for most problems is kind of okay.

But you can avoid it if you want to get the blazing fast solution.

But now let's actually go ahead and implement my favorite version, which would be a recursive version, right, which is technically of the depth first.

Right? So we'll say const, trimming value, chattering min value, dream and value.

And to establish this one recursively will always start with our base case, right? That's when you saw these curly guys.

So I'll start my base case, I'll say if my routes is equal to null, then that kind of means I have the empty tree, what's like the minimum value in the empty tree? While that must be infinity, right? For the same reason we chose infinity before, right? It's just a good default value, because I'm going to minimize in the long run.

Cool.

And now what I want to do is I want to make my recursive call.

So if I call treeman value on my left child, and I call it on my right child, I'll just think about what these function calls return.

Right? Although they're recursive, if these calls work, they should give me back the smallest value in the left subtree.

And the smallest value in the right subtree.

So maybe, just to be clear, I'll go ahead and save the system variables.

So I'll say const, I'll say, left.

Min.

And also this should be the right men, right? So I found the smallest in the left and the smallest in the right.

But I also need to think about myself, right? I am ru dot Val.

So I need to choose the smallest of root Val left men and Reitman, right, so you can write like a conditional.

So just some if statements If you want, I think the best way to implement this in JavaScript is just to use the math dot min function.

So JavaScript, you always have access to the global capital math object has a few useful methods on it.

One we're going to find useful right now is math dot min.

If you do that, you can pass in some arguments and an arbitrary number of arguments.

So give me the smallest between route dot Val and be sure to access the value because you need to give numbers to men, right? Smaller between route dot Val, my left men and my right man.

Cool.

And this feels somewhat similar to the approach drawn we did right where we had to choose the smallest of these three numbers.

So before we run it, maybe you're unfamiliar with math dot min, I can quickly demo that for you.

Let's say I had some numbers, we had 10, we had three, we had six, and we had negative 12.

And also 100.

And if I console log with that returns, I'll just run this file as a script.

So I'm not going to execute the tests quite yet.

So I should get negative 12.

Because that's the smallest perfect.

And now I think we're ready to go ahead and test this.

So our recursive version, there we have it.

Cool.

Alright, so there, we have our JavaScript walkthrough for this tree minvalue problem, recommend you practice all of these different solutions, and then pick a favorite, maybe just have in your back pocket.

That way, you can always solve this problem in the future, right.

So you should have no issue thinking about how to solve like a tree maxvalue problem.

And by now, you know, we're getting pretty comfortable with trees.

And typically, we're always going to fall back to either a breadth first or a depth first traversal.

But in the next few problems, we're definitely going to look at some spin offs and some variations on how we can accomplish more complex logic using these algorithms as our baseline.

So I'll catch in the next one.

Hey, programmers, Alvin here, right now want to go over this max root to leaf path, some problem, pretty wordy title, let's take a look at what this problem asks us to do.

So we're going to be taking in a binary tree as input, and we want to do is really talk about the paths within this tree in particular, root to leaf paths.

So just to review, we know that a binary tree typically for us has a single root and you can identify the root by just looking at the topmost node, that is the node with no parent, right? And then also given this tree there are three leaves, right? a leaf is a node that has no children.

So do bear in mind right a root has no parent.

A leaf node has no children, some Like 11, and three in this picture, there are neither roots nor leaves, right.

So it could be the case that a tree has many leaves, like this tree, right has three leaves.

But a tree for us typically only has a single root.

And so what we want to do is consider all three of the different root to leave paths, right, so you must start at the root, and you must terminate at a leaf.

So one of those paths would just be this one, right 511 for, what we want to do is consider its total sum.

So just summing the values within the nodes along that path.

If I do five plus 11, plus four, that would give me a total sum of 20.

So that's just one of the possibilities.

If I look at another path, let's say this path of 511 to five plus 11, plus two would give me 18.

That's another option.

The only other option is is final path from five to one, five to five plus three plus one.

That gives me a total sum of nine, right? And now among those three options, I want to choose the maximal path some right? So that would be a final answer of 20.

So in the long run, we need to come up with some code that computes the maximal path sum from the root to any leaf.

And that sounds like a pretty hard problem, right? How can we go about doing that one? Well, hopefully you notice some similar patterns, right to problems we've done previously.

So just by looking at the name of this problem, I'm reminded of a few different patterns, right? I can think of the tree some problem seems pretty related here.

And can I also think about, like that minvalue.

Problem are, we had some minimization logic, except now, I want to maximize something here.

And so probably what I'll have to do in this one is combined some of my previous knowledge to come up with a pretty novel solution, but it's okay to rely heavily on our previous experiences, right.

And so let's start with something classic, I think we're gonna solve this one recursively.

And in the grand scheme of things, typically, for tree problems, that is, a recursive code is usually your best bet when it comes to like, pathfinding things, right, and building pads and determining pads.

And so we'll take a look at what we should have our base cases be, if we're gonna frame this one recursively.

So they tell us that, all right, you need to consider paths, but not just any paths, right? You always want to end at a leaf node.

So the leaf nodes, like the ending point, belief, notice the base case.

So my base case is going to be literally about a leaf node.

And recall that that's a node with no children.

So for example, let's say we had a node as input that had no children, then what's the total sum of that leaf node? Well would just be its inner value n, right? So kind of think about your base cases as if they're their own inputs, right? As a quick aside, right, so a totally separate scenario, right? Now, let's say I gave you a tree that contains only one node, and its value was 42.

And I asked you to, alright, tell me, what's the maximum path some of this tree, this tree is very small, right? If you identify the root in this tree, it's just the 42 node, right? Because it has no parent.

And I've also asked you for the leaves, there's only one leaf here.

And it's also the 42, right? Because it has no children, right? So the single note of 42, in this example, is the root and the leaf, which means that the maximum path, some would just be 42.

Cool.

So I'm thinking about my base cases as if they're their own inputs.

Nice.

And what you'd probably recognize here is, in this scenario, we're kind of considering a base case, that is not about a null node, right, we will need at least a separate base case that checks if we have an actual leaf node.

And that's very, very inherent and given in the problem, right? They say root to leaf path.

Awesome.

So if that's the case, I'm just going to locate all of my leaf nodes in this diagram.

And I know that they're going to return the values within them.

So just plugging those in, right? And now I can start reconstructing on my higher level solutions, right? So given this 11 node, what decision does this 11 node have to make to compute its answer, right? So let's read ourselves at this 11 node.

And we have to consider our left and right values, right.

And so if i route myself at 11, the four represents my max path sum through my left subtree.

And the right hand side to now represents the max path sum through my right subtree.

Since I want to maximize here, the key is to just choose between four and two, right? I choose the bigger of them, right? So four is bigger than two, so I must use the four.

And what I do is I take that four, and I add my current value of 11 into it, and that gives me 15.

And that would actually be the answer for this subtree.

In other words, I can check for correctness right now, if I root myself at 11.

I'm returning 15 to represent this path, right? I can either do 11 plus four, which is 15.

Or I can do 11 plus two, which is 13.

Right and that 13 is smaller, so I won't prefer it here.

Cool.

So so far, it seems like we're in good shape.

Now let's take a look at another node read to evaluate.

Let's look at this.

Three node.

So it has a value on its right hand side, right? It's receiving a one from its right child, but the left child doesn't exist, right? And if you want to be a little more explicit, we know that the left child or three dot left is going to be a null pointer.

All right? So kind of plug that in here.

And here's where we should work in possibly another base case, sometimes it's very natural to course correct as we go.

And so for a null node, what should we return? Well, we know that whenever we have a null node, it should never contribute to our final answer, we can kind of just ignore it, I guess, we want to make sure that it's compatible with the rest of our internal logic.

So bear in mind in this problem, I want to take the maximum right? Recall from our previous problem, right, the min value problem, we want to take the minimum.

And so we made our null nodes return infinity, right? Because if I want to find the minimum, and my null nodes, return infinity, by default, I know that infinity is never going to win a comparison, right? Because infinity is very large.

And we just want to flip that logic over here, right? So because in the long run, I want to find the maximum, I'll make my base case for the null nodes return negative infinity, right? Because I know negative infinity is never going to win any contest where we compare things, looking for the larger of the values, right? So for this null note, I'll plug in that negative infinity.

And now I shouldn't be able to do the same business logic as before, right? So I'm rooting myself at the three node, and I can look at the results I get for my children, right.

So between my two children in my left path, I get a negative infinity, or in my right path, I get a one, and I choose the maximum between them, right? One is bigger than negative infinity.

So I should use the one, right.

And what I do with that one is, I add my current value of three to it.

So three plus one gives me four, right.

And that makes some sense, because if I look at that small subtree, rooted in three, the biggest path from root to leaf, or sub root to leaf, I would indeed be a four.

Nice, and now we have ourselves at the ultimate root over here, I do that similar comparison, right? I check the bigger of my two children, so I'm going to keep the 15.

But then I add myself to it giving 20.

Like we said before, this 20 logically represents this path of five plus 11 plus four.

Cool, so there we have it.

And if you want to take a look at the complexity of this seems nothing unusual.

So we're going to have n as the number of nodes, we're going to have O of n runtime, because we're going to have to make a call a recursive call that is for every node within the tree.

And if I think about any particular call, we're going to make a foreign node, we're just going to do a comparison, right, I just choose the bigger of my two children.

So we're not going to have any, any loops within our calls, I believe.

Cool.

So it just seems to be O of n time.

Like we always say space complexity is O of n, just due to the call stack.

Nice.

So let's go ahead and code this one up.

And it should feel like a combination of our previous tree some problems, as well as some a min value max value logic.

Hey, programmers, welcome back, Ryan, I want to go over a JavaScript solution for this max root to leaf path, some problem.

So this is going to be a pretty interesting tree problem.

And we'll jump right into the code, please make sure you watch the approach video before this.

So I think should be all primed up with the logic that we want to implement here.

Right? And so we'll start by doing this recursively.

And I'd say that's probably the best way to solve this one.

So I think it's the only way I'll show you, when I want to do is start with a base case, right? So we said we want to figure out our root to leaf paths.

So our base case should be about whether I have a leaf, right? So I'm going to go ahead and check let's say, if my root left is equal to No, right? And root, right? Is equal to No, right? That's the definition of a leaf.

So if I have the left child and I have the right child, then I should return the value stored at this node, right? So root, dot Val.

Nice.

So again, we're kind of emphasizing here is that base case catches a scenario.

If we're at like four, right, we have no children.

Nice.

And we will need another base case in the long run.

For now, let's say we left it like this, right? So we said that in the context of solving this one recursively, the decision we make at every note is I choose the bigger result from my left column A right call, and then I add myself to it, right? So I know I'm going to do this recursively so I'm going to call the same function.

And right now you should be familiar with all right, you call your function on root left, and also root right? That I think about what those calls return, right? They give me back, the max pass some through my left subtree on the max path, some through my right subtree I just want to choose the bigger results between Windows two.

So no I swear to you saw me use the math dot min function here I'll use the math dot max function.

And I know that these calls these expressions evaluates to numbers.

So I can actually just pass these guys in, in line.

So I would just move this up like that.

So when finding the max between these two results, and I'll go ahead and call that we'll say, the max, your child, right? Maybe I'll write this online, they prefer.

And then from there, I just add myself to that max child, right? So I'm going to return route dot valance myself, plus max child.

And we'll call that max child path.

And that should be it except for one thing we're forgetting.

So take a moment, see if you can read between the lines here.

And what scenario Am I missing? We did mention it in the approach video.

So can you spot what is not present here? And that's something I find I do often right? If I'm able to really come up with a nice, meaningful and consistent picture, then everything I drew in the picture should translate to some piece of code over here.

Let's give it a shot.

Expect to fail this one.

See what we get? Yeah, so we're getting an error cannot read property left have no, right, I found on the very first example.

So I'll tell you that this is caused by an asymmetric node, like this four over here, right? This four has no left child.

So imagine what would happen in this instance, right? So if we're evaluating this code, we know that our route is going to be this four node.

And this if statement on line 10, is not hit, right? Because four is not a leaf, right? Because it has a right child's right child does not know.

So instead, it's going to run until 11.

And it's going to try to execute max path, some of root dot left root that left is no.

And so when we actually evaluate that call, now we're seeing is no, and we're gonna check no dot left over here.

And that's going to break right? It's exactly the line that's broken.

And so in addition to this base case, right, when we bought him out at a leaf node, what if we just end up at a null node, so if root is null, then we said that a good value to return would be negative infinity, right? Because I'm choosing to, in the long run, do like maximal logic and want to pick the max.

So if I choose a negative infinity, that won't interfere with any of us taking a max, right? Because imagine that this thing was negative infinity.

And I would just prefer the right hand side if it was not infinity, right? Negative infinity that is.

And so with that, let's go ahead and give this a go.

Cool.

And then we have a nice solution for our max root to leaf path some problem.

So notice how short the code is, by no means do I think this is simple code, though, right? At this point, it's okay if it feels a little tough.

But you understand how we have some familiar patterns, right, I still have those two recursive calls right over here.

And we typically only vary in how we take those recursive calls or the results from them, and combine them into our higher level answer.

So keep practicing this problem and I'll catch you in the next one.

All right programmers.

That wraps up our course on the introduction to binary trees.

If you want to explore this binary tree topic, or really any other data structures and algorithms topic more deeply.

Be sure to head destructing dotnet where we have all of these topics covered through tons of problems where we have video walkthroughs and illustrations for every single problem.

I think you'll find this especially useful if you're repairing and really cramming for those technical interviews.

Thanks for watching, and I'll see you there.