A tree is a data structure composed of nodes that has the following characteristics:

1. Each tree has a root node (at the top) having some value.
2. The root node has zero or more child nodes.
3. Each child node has zero or more child nodes, and so on. This create a subtree in the tree. Every node has it’s own subtree made up of his children and their children, etc. This means that every node on its own can be a tree.

A binary search tree (BST) adds these two characteristics:

1. Each node has a maximum of up to two children.
2. For each node, the values of its left descendent nodes are less than that of the current node, which in turn is less than the right descendent nodes (if any).

The BST is built up on the idea of the binary search algorithm, which allows for fast lookup, insertion and removal of nodes. The way that they are set up means that, on average, each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the logarithm of the number of items stored in the tree, `O(log n)`.

However, some times the worst case can happen, when the tree isn’t balanced and the time complexity is `O(n)` for all three of these functions. That is why self-balancing trees (AVL, red-black, etc.) are a lot more effective than the basic BST.

Worst case scenario example: This can happen when you keep adding nodes that are always larger than the node before (it’s parent), the same can happen when you always add nodes with values lower than their parents.

## Basic operations on a BST

• Create: creates an empty tree.
• Insert: insert a node in the tree.
• Search: Searches for a node in the tree.
• Delete: deletes a node from the tree.

### Create

Initially an empty tree without any nodes is created. The variable/identifier which must point to the root node is initialized with a `NULL` value.

You always start searching the tree at the root node and go down from there. You compare the data in each node with the one you are looking for. If the compared node doesn’t match then you either proceed to the right child or the left child, which depends on the outcome of the following comparison: If the node that you are searching for is lower than the one you were comparing it with, you proceed to to the left child, otherwise (if it’s larger) you go to the right child. Why? Because the BST is structured (as per its definition), that the right child is always larger than the parent and the left child is always lesser.

### Insert

It is very similar to the search function. You again start at the root of the tree and go down recursively, searching for the right place to insert our new node, in the same way as explained in the search function. If a node with the same value is already in the tree, you can choose to either insert the duplicate or not. Some trees allow duplicates, some don’t. It depends on the certain implementation.

### Delete

There are 3 cases that can happen when you are trying to delete a node. If it has,

1. No subtree (no children): This one is the easiest one. You can simply just delete the node, without any additional actions required.
2. One subtree (one child): You have to make sure that after the node is deleted, its child is then connected to the deleted node’s parent.
3. Two subtrees (two children): You have to find and replace the node you want to delete with its successor (the letfmost node in the right subtree).

The time complexity for creating a tree is `O(1)`. The time complexity for searching, inserting or deleting a node depends on the height of the tree `h`, so the worst case is `O(h)`.

### Predecessor of a node

Predecessors can be described as the node that would come right before the node you are currently at. To find the predecessor of the current node, look at the right-most/largest leaf node in the left subtree.

### Successor of a node

Successors can be described as the node that would come right after the node you are currently at. To find the successor of the current node, look at the left-most/smallest leaf node in the right subtree.

## Special types of BT

• Heap
• Red-black tree
• B-tree
• Splay tree
• N-ary tree

## Runtime

### Data structure: Array

• Worst-case performance: `O(log n)`
• Best-case performance: `O(1)`
• Average performance: `O(log n)`
• Worst-case space complexity: `O(1)`

Where `n` is the number of nodes in the BST.

## Implementation of BST

Here’s a definiton for a BST node having some data, referencing to its left and right child nodes.

``````struct node {
int data;
struct node *leftChild;
struct node *rightChild;
};``````

### Search Operation

Whenever an element is to be searched, start searching from the root node. Then if the data is less than the key value, search for the element in the left subtree. Otherwise, search for the element in the right subtree. Follow the same algorithm for each node.

``````struct node* search(int data){
struct node *current = root;
printf("Visiting elements: ");

while(current->data != data){

if(current != NULL) {
printf("%d ",current->data);

//go to left tree
if(current->data > data){
current = current->leftChild;
}//else go to right tree
else {
current = current->rightChild;
}

if(current == NULL){
return NULL;
}
}
}
return current;
}``````

### Insert Operation

Whenever an element is to be inserted, first locate its proper location. Start searching from the root node, then if the data is less than the key value, search for the empty location in the left subtree and insert the data. Otherwise, search for the empty location in the right subtree and insert the data.

``````void insert(int data) {
struct node *tempNode = (struct node*) malloc(sizeof(struct node));
struct node *current;
struct node *parent;

tempNode->data = data;
tempNode->leftChild = NULL;
tempNode->rightChild = NULL;

//if tree is empty
if(root == NULL) {
root = tempNode;
} else {
current = root;
parent = NULL;

while(1) {
parent = current;

//go to left of the tree
if(data < parent->data) {
current = current->leftChild;
//insert to the left

if(current == NULL) {
parent->leftChild = tempNode;
return;
}
}//go to right of the tree
else {
current = current->rightChild;

//insert to the right
if(current == NULL) {
parent->rightChild = tempNode;
return;
}
}
}
}
}        ``````

Binary search trees (BSTs) also give us quick access to predecessors and successors. Predecessors can be described as the node that would come right before the node you are currently at.

• To find the predecessor of the current node, look at the rightmost/largest leaf node in the left subtree. Successors can be described as the node that would come right after the node you are currently at.
• To find the successor of the current node, look at the leftmost/smallest leaf node in the right subtree.

## Let’s look at a couple of procedures operating on trees.

Since trees are recursively defined, it’s very common to write routines that operate on trees that are themselves recursive.

So for instance, if we want to calculate the height of a tree, that is the height of a root node, We can go ahead and recursively do that, going through the tree. So we can say:

• For instance, if we have a nil tree, then its height is a 0.
• Otherwise, We’re 1 plus the maximum of the left child tree and the right child tree.

So if we look at a leaf for example, that height would be 1 because the height of the left child is nil, is 0, and the height of the nil right child is also 0. So the max of that is 0, then 1 plus 0.

### Height(tree) algorithm

``````if tree = nil:
return 0
return 1 + Max(Height(tree.left),Height(tree.right))``````

### Here is the code in C++

``````int maxDepth(struct node* node)
{
if (node==NULL)
return 0;
else
{
int rDepth = maxDepth(node->right);
int lDepth = maxDepth(node->left);

if (lDepth > rDepth)
{
return(lDepth+1);
}
else
{
return(rDepth+1);
}
}
}  ``````

We could also look at calculating the size of a tree that is the number of nodes.

• Again, if we have a nil tree, we have zero nodes.

Otherwise, we have the number of nodes in the left child plus 1 for ourselves plus the number of nodes in the right child. So 1 plus the size of the left tree plus the size of the right tree.

### Size(tree) algorithm

``````if tree = nil
return 0
return 1 + Size(tree.left) + Size(tree.right)``````

### Here is the code in C++

``````int treeSize(struct node* node)
{
if (node==NULL)
return 0;
else
return 1+(treeSize(node->left) + treeSize(node->right));
}``````

## Following are common types of Binary Trees:

Full Binary Tree/Strict Binary Tree: A Binary Tree is full or strict if every node has exactly 0 or 2 children.

``````           18
/       \
15         30
/  \        /  \
40    50    100   40``````

In Full Binary Tree, number of leaf nodes is equal to number of internal nodes plus one.

Complete Binary Tree: A Binary Tree is complete Binary Tree if all levels are completely filled except possibly the last level and the last level has all keys as left as possible

``````           18
/       \
15         30
/  \        /  \
40    50    100   40
/  \   /
8   7  9 ``````