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**Binary Search Tree** is a node-based binary tree data structure which has the following properties:

- The left subtree of a node contains only nodes with keys lesser than the node’s key.
- The right subtree of a node contains only nodes with keys greater than the node’s key.
- The left and right subtree each must also be a binary search tree.

Below is the Example of Binary Search Tree.

**Background:** The worst case time complexity of search and insert operations is O(h) where h is height of Binary Search Tree. In worst case, we may have to travel from root to the deepest leaf node. The height of a skewed tree may become n and the time complexity of search and insert operation may become O(n).

## Algorithm for finding minimum or maximum element in Binary Search Tree

As we know the Property of Binary search tree.This quite simple

**Approch for finding minimum element:**

- Traverse the node from root to left recursively until left is NULL.
- The node whose left is NULL is the node with minimum value.

**Approch for finding maximum element:**

- Traverse the node from root to right recursively until right is NULL.
- The node whose right is NULL is the node with maximum value.

Implementation of the above approches.

```
// C++ program to find maximum or minimum element in binary search tree
#include <bits/stdc++.h>
using namespace std;
struct node
{
int key;
struct node *left, *right;
};
// A utility function to create a new BST node
struct node *newNode(int item)
{
struct node *temp = (struct node *)malloc(sizeof(struct node));
temp->key = item;
temp->left = temp->right = NULL;
return temp;
}
/* A utility function to insert a new node with given key in BST */
struct node* insert(struct node* node, int key)
{
struct node *newNode(int );
/* If the tree is empty, return a new node */
if (node == NULL) return newNode(key);
/* Otherwise, recur down the tree */
if (key < node->key)
node->left = insert(node->left, key);
else if (key > node->key)
node->right = insert(node->right, key);
/* return the (unchanged) node pointer */
return node;
}
/* Given a non-empty binary search tree,
return the minimum data value found in that
tree. Note that the entire tree does not need
to be searched. */
int minValue(struct node* node)
{
struct node* current = node;
/* loop down to find the leftmost leaf */
while (current->left != NULL)
{
current = current->left;
}
return(current->key);
}
/* Given a non-empty binary search tree,
return the maximum data value found in that
tree. Note that the entire tree does not need
to be searched. */
int maxValue(struct node* node)
{
struct node* current = node;
/* loop down to find the leftmost leaf */
while (current->right != NULL)
{
current = current->right;
}
return(current->key);
}
// Driver Program to test above functions
int main()
{
int maxValue(struct node* );
struct node* insert(struct node* , int );
int minValue(struct node* );
struct node *root = NULL;
root = insert(root, 8);
insert(root, 3);
insert(root, 10);
insert(root, 1);
insert(root, 6);
insert(root, 4);
insert(root, 7);
insert(root, 5);
insert(root, 10);
insert(root, 9);
insert(root, 13);
insert(root, 11);
insert(root, 14);
insert(root, 12);
insert(root, 2);
cout << "\n Minimum value in BST is " << minValue(root)<<endl;
cout << "\n Maximum value in BST is " << maxValue(root);
return 0;
}
```

Output:

```
Minimum value in BST is 1
Maximum value in BST is 14
```

## Explanation:

For Finding Minimum value in Binary search tree.

- start from root i.e 8.

- As left of root is not null go to left of root i.e 3.

- As left of 3 is not null go to left of 3 i.e. 1.
- Now as the left of 1 is null therefore 1 is the minimum element

For Finding Maximum value in Binary search tree. - start from root i.e 8.

- As right of root is not null go to right of root i.e 10.

- As right of 10 is not null go to right of root i.e 13.

- As right of 13 is not null go to right of root i.e 14.
- Now as the right of 14 is null therefore 14 is the maximum element.

## Time Complexity:

**O(N)** Worst case happens for left skewed trees in finding the minimum value.

**O(N)** Worst case happens for right skewed trees in finding the maximum value.

**O(1)** Best case happens for left skewed trees in finding the maximum value.

**O(1)** Best case happens for right skewed trees in finding the minimum value.

In the balanced Binary Search tree, the time complexity is **O(log N)**

N is the number of elements in a Binary Search tree