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# Binary Search Tree: Insertion

## Insertion in a Binary Search Tree

**Binary Search tree is fairly easy and we just need to know a few simple rules given below –**

**Rules for Insertion in a Binary Search Tree (BST)**

- The
**left subtree**for any given node will only contain nodes which are lesser than the current node - The
**right subtree**for any given node will only contain nodes which are greater than the current node - Each subtree must also follow the above rules of BST

### Pre Requisite to Insertion

Pre-Requisite
To insert in Binary search tree we need to search for the correct node where the new node should be insert

So, first, we will learn how to search a given node(value), in a binary search tree

## Searching in a Binary Search Tree

Let us say we want to 9 in the binary search tree given in our example above, what approach we would follow?

### Algorithm

- Start from the root node
- Compare the value with the root node, if same then found, return True
- If the value is less than the root node, recurse to the left node. Else recurse to the right node
- Do this recursively, until the value is found or not found

### Example

Assuming we want to find 9 in the above example

- The root node is 15, (9 <15) thus recursing to the left node
- Now, the left node is 13, (9<14) thus again, recursing to the left node
- Now, the left node is 8, (9>8) thus, recursing to the right node
- Finally, the right node is 9, (9=9), thus value found, return true

### Code to implement Search

// A sample C function to check if a given node exists in a binary search tree or not

struct node* search(struct node* root, int value)

{

// Check if the root is null or value is present at root itself

if (root == NULL || root->value == value)

return root;

// Root is smaller than value, go to right subtree's node, doing recusion here

if (root->value < value)

return search(root->right, value);

// else only one case is left

// i.e. Root is greater than value, go to left subtree's node, doing recusion here

return search(root->left, value);

}

## Insertion in a Binary Search Tree

Insertion is fairly simple if you understood search perfectly

- Try to search the BST for the node to be inserted
- As the node wouldn’t exist we will hit the leaf node, with null as child node right!
- Insert the new node here
- Voila! Party, you’ve inserted

### Code for Binary Search Tree Insertion

// PrepInsta's program to do insertion in a Binary Search Tree (BST)

#include<stdio.h>

#include<stdlib.h>

// Basic struct of Tree

struct node

{

int val;

struct node *left, *right;

};

// Function to create a new Node

struct node* newNode(int item)

{

struct node* temp = (struct node *)malloc(sizeof(struct node));

temp->val = item;

temp->left = temp->right = NULL;

return temp;

}

// Function print the node in inorder format, when insertion is complete

void inorder(struct node* root)

{

if (root != NULL)

{

inorder(root->left);

printf("%d \n", root->val);

inorder(root->right);

}

}

// Here we are finding where to insert the new node so BST is followed

struct node* insert(struct node* node, int val)

{

/* If the tree(subtree) is empty, return a new node by calling newNode func.*/

if (node == NULL) return newNode(val);

/* Else, we will do recursion down the tree to further subtrees */

if (val < node->val)

node->left = insert(node->left, val);

else if (val > node->val)

node->right = insert(node->right, val);

/* (Safety) return the node's pointer which is unchanged */

return node;

}

int main()

{

/* Our BST will look like this

100

/ \

40 140

/ \ / \

40 80 120 160 */

struct node* root = NULL;

root = insert(root, 100);

insert(root, 60);

insert(root, 40);

insert(root, 80);

insert(root, 140);

insert(root, 120);

insert(root, 160);

// Finally printing the tree using inorder

inorder(root);

return 0;

}

### How it works?

- In binary search trees, we use the INSERT function to add a new element in a tree.
- Insertion is similar to searching wherein, we first check if the element is present in the given data or not. If not, the node is entered at that position.
- If the data to be added is greater than the parent data node it is inserted towards the right of the parent; else it is inserted to the left of the parent.
- In other words,

- If
*root > node,*

then the data node is added to the *left sub-tree.*

- If
*root < node,*

then the data node is added to the *right sub-tree.*

For Example: **[15, 18, 13, 8, 14, 4, 16, 19, 9]**

**15 being the first element becomes the root of the tree.**

- The next element,
**18**, is now compared against the root node. Since**18 > 15**it is inserted to the right hand side of the root node.

- The next element,
**13**, is compared to the root node. Since**15 > 13**it is inserted to the left of the root node.

- Now the next element
**8**is compared to the root node. Since**8 < 15**it has to be inserted to the left subtree. Now again**8**is compared to**13**and since**8 < 13**it is added to the left of 13.

- The next element to be inserted is
**14**. Since**14****< 15**it will be added to the left subtree. Now again**14**is compared to**13**and since**14 > 13**it is added to the right of 13.

- The next element to be inserted is
**4**. Since**4 < 15**it is to be added to the left subtree. Now again**4**compared to**13**and then it is compared to**8**and since**4 < 13**and**4 < 8**it is added to the left of 8.

- The next element to be inserted is
**16**. Since**16 > 15**it is to be added to the right subtree. Now again**16**is compared to**18**and since**16 < 18**it is added to the left of 18.

- The next element to be inserted is
**19**. Since**19 > 15**it is to be added to the right subtree. Now again**19**is compared to**18**and since**19 > 18**it is added to the right of 19.

- The last element to be inserted is
**9**. Since**9 < 15**it is to be added to the left sub-tree. Now again**9**is compared to**13**and is then compared**8**, and since**9 < 13**but**9 > 8**it is added to the right of 8.

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