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Find median of BST

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  • Difficulty Level : Hard
  • Last Updated : 01 Sep, 2022
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Given a Binary Search Tree, find the median of it. 

  • If number of nodes are even: then median = ((n/2th node + ((n)/2th+1) node) /2 
  • If number of nodes are odd: then median = (n+1)/2th node.

Examples:

Input:

Example of BST

Example of BST

Output: median of Below BST is 6.
Explanation: Inorder of Given BST will be  
1, 3, 4, 6, 7, 8, 9 So, here median will 6.

Given BST:
 

Output: median of Below BST is 12

Median Of BST using Morris Inorder Traversal:

The idea is based on K’th smallest element in BST using O(1) Extra Space.

The task is very simple if we are allowed to use extra space but Inorder to traversal using recursion and stack both use Space which is not allowed here. So, the solution is to do Morris Inorder traversal as it doesn’t require extra space.

Follow the steps mentioned below to implement the idea:

  • Count the number of nodes in the given BST using Morris Inorder Traversal.
  • Then perform Morris Inorder traversal one more time by counting nodes and by checking if the count is equal to the median point.
  • To consider even no. of nodes, an extra pointer pointing to the previous node is used.

Below is the implementation of the above approach:

C++




/* C++ program to find the median of BST in O(n)
   time and O(1) space*/
#include<bits/stdc++.h>
using namespace std;
 
/* A binary search tree Node has data, pointer
   to left child and a pointer to right child */
struct Node
{
    int data;
    struct Node* left, *right;
};
 
// A utility function to create a new BST node
struct Node *newNode(int item)
{
    struct Node *temp =  new Node;
    temp->data = 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)
{
    /* If the tree is empty, return a new node */
    if (node == NULL) return newNode(key);
 
    /* Otherwise, recur down the tree */
    if (key < node->data)
        node->left  = insert(node->left, key);
    else if (key > node->data)
        node->right = insert(node->right, key);
 
    /* return the (unchanged) node pointer */
    return node;
}
 
/* Function to count nodes in a  binary search tree
   using Morris Inorder traversal*/
int counNodes(struct Node *root)
{
    struct Node *current, *pre;
 
    // Initialise count of nodes as 0
    int count = 0;
 
    if (root == NULL)
        return count;
 
    current = root;
    while (current != NULL)
    {
        if (current->left == NULL)
        {
            // Count node if its left is NULL
            count++;
 
            // Move to its right
            current = current->right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current->left;
 
            while (pre->right != NULL &&
                   pre->right != current)
                pre = pre->right;
 
            /* Make current as right child of its
               inorder predecessor */
            if(pre->right == NULL)
            {
                pre->right = current;
                current = current->left;
            }
 
            /* Revert the changes made in if part to
               restore the original tree i.e., fix
               the right child of predecessor */
            else
            {
                pre->right = NULL;
 
                // Increment count if the current
                // node is to be visited
                count++;
                current = current->right;
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
 
    return count;
}
 
 
/* Function to find median in O(n) time and O(1) space
   using Morris Inorder traversal*/
int findMedian(struct Node *root)
{
   if (root == NULL)
        return 0;
 
    int count = counNodes(root);
    int currCount = 0;
    struct Node *current = root, *pre, *prev;
 
    while (current != NULL)
    {
        if (current->left == NULL)
        {
            // count current node
            currCount++;
 
            // check if current node is the median
            // Odd case
            if (count % 2 != 0 && currCount == (count+1)/2)
                return current->data;
 
            // Even case
            else if (count % 2 == 0 && currCount == (count/2)+1)
                return (prev->data + current->data)/2;
 
            // Update prev for even no. of nodes
            prev = current;
 
            //Move to the right
            current = current->right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current->left;
            while (pre->right != NULL && pre->right != current)
                pre = pre->right;
 
            /* Make current as right child of its inorder predecessor */
            if (pre->right == NULL)
            {
                pre->right = current;
                current = current->left;
            }
 
            /* Revert the changes made in if part to restore the original
              tree i.e., fix the right child of predecessor */
            else
            {
                pre->right = NULL;
 
                prev = pre;
 
                // Count current node
                currCount++;
 
                // Check if the current node is the median
                if (count % 2 != 0 && currCount == (count+1)/2 )
                    return current->data;
 
                else if (count%2==0 && currCount == (count/2)+1)
                    return (prev->data+current->data)/2;
 
                // update prev node for the case of even
                // no. of nodes
                prev = current;
                current = current->right;
 
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
}
 
/* Driver program to test above functions*/
int main()
{
 
    /* Let us create following BST
                  50
               /     \
              30      70
             /  \    /  \
           20   40  60   80 */
    struct Node *root = NULL;
    root = insert(root, 50);
    insert(root, 30);
    insert(root, 20);
    insert(root, 40);
    insert(root, 70);
    insert(root, 60);
    insert(root, 80);
 
    cout << "\nMedian of BST is "
         << findMedian(root);
    return 0;
}

Java




/* Java program to find the median of BST in O(n)
time and O(1) space*/
class GfG {
 
/* A binary search tree Node has data, pointer
to left child and a pointer to right child */
static class Node
{
    int data;
    Node left, right;
}
 
// A utility function to create a new BST node
static Node newNode(int item)
{
    Node temp = new Node();
    temp.data = item;
    temp.left = null;
    temp.right = null;
    return temp;
}
 
/* A utility function to insert a new node with
given key in BST */
static Node insert(Node node, int key)
{
    /* If the tree is empty, return a new node */
    if (node == null) return newNode(key);
 
    /* Otherwise, recur down the tree */
    if (key < node.data)
        node.left = insert(node.left, key);
    else if (key > node.data)
        node.right = insert(node.right, key);
 
    /* return the (unchanged) node pointer */
    return node;
}
 
/* Function to count nodes in a binary search tree
using Morris Inorder traversal*/
static int counNodes(Node root)
{
    Node current, pre;
 
    // Initialise count of nodes as 0
    int count = 0;
 
    if (root == null)
        return count;
 
    current = root;
    while (current != null)
    {
        if (current.left == null)
        {
            // Count node if its left is NULL
            count++;
 
            // Move to its right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
 
            while (pre.right != null &&
                pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its
            inorder predecessor */
            if(pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if part to
            restore the original tree i.e., fix
            the right child of predecessor */
            else
            {
                pre.right = null;
 
                // Increment count if the current
                // node is to be visited
                count++;
                current = current.right;
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
 
    return count;
}
 
 
/* Function to find median in O(n) time and O(1) space
using Morris Inorder traversal*/
static int findMedian(Node root)
{
if (root == null)
        return 0;
 
    int count = counNodes(root);
    int currCount = 0;
    Node current = root, pre = null, prev = null;
 
    while (current != null)
    {
        if (current.left == null)
        {
            // count current node
            currCount++;
 
            // check if current node is the median
            // Odd case
            if (count % 2 != 0 && currCount == (count+1)/2)
                return prev.data;
 
            // Even case
            else if (count % 2 == 0 && currCount == (count/2)+1)
                return (prev.data + current.data)/2;
 
            // Update prev for even no. of nodes
            prev = current;
 
            //Move to the right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
            while (pre.right != null && pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its inorder predecessor */
            if (pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if part to restore the original
            tree i.e., fix the right child of predecessor */
            else
            {
                pre.right = null;
 
                prev = pre;
 
                // Count current node
                currCount++;
 
                // Check if the current node is the median
                if (count % 2 != 0 && currCount == (count+1)/2 )
                    return current.data;
 
                else if (count%2==0 && currCount == (count/2)+1)
                    return (prev.data+current.data)/2;
 
                // update prev node for the case of even
                // no. of nodes
                prev = current;
                current = current.right;
 
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
    return -1;
}
 
/* Driver code*/
public static void main(String[] args)
{
 
    /* Let us create following BST
                50
            / \
            30 70
            / \ / \
        20 40 60 80 */
    Node root = null;
    root = insert(root, 50);
    insert(root, 30);
    insert(root, 20);
    insert(root, 40);
    insert(root, 70);
    insert(root, 60);
    insert(root, 80);
 
    System.out.println("Median of BST is " + findMedian(root));
}
}
 
// This code is contributed by prerna saini.

Python3




# Python program to find closest
# value in Binary search Tree
 
_MIN=-2147483648
_MAX=2147483648
 
# Helper function that allocates 
# a new node with the given data 
# and None left and right pointers.                                    
class newNode:
 
    # Constructor to create a new node
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None
 
""" A utility function to insert a new node with
given key in BST """
def insert(node,key):
 
    """ If the tree is empty, return a new node """
    if (node == None):
        return newNode(key)
 
    """ Otherwise, recur down the tree """
    if (key < node.data):
        node.left = insert(node.left, key)
    elif (key > node.data):
        node.right = insert(node.right, key)
 
    """ return the (unchanged) node pointer """
    return node
 
 
""" Function to count nodes in
    a binary search tree using
     Morris Inorder traversal"""
def counNodes(root):
 
    # Initialise count of nodes as 0
    count = 0
 
    if (root == None):
        return count
 
    current = root
    while (current != None):
     
        if (current.left == None):
         
            # Count node if its left is None
            count+=1
 
            # Move to its right
            current = current.right
         
        else:    
            """ Find the inorder predecessor of current """
            pre = current.left
 
            while (pre.right != None and
                    pre.right != current):
                pre = pre.right
 
            """ Make current as right child of its
            inorder predecessor """
            if(pre.right == None):
             
                pre.right = current
                current = current.left
            else:
             
                pre.right = None
 
                # Increment count if the current
                # node is to be visited
                count += 1
                current = current.right
 
    return count
 
 
 
""" Function to find median in
    O(n) time and O(1) space
    using Morris Inorder traversal"""
def findMedian(root):
    if (root == None):
        return 0
    count = counNodes(root)
    currCount = 0
    current = root
 
    while (current != None):
     
        if (current.left == None):
         
            # count current node
            currCount += 1
 
            # check if current node is the median
            # Odd case
            if (count % 2 != 0 and
                currCount == (count + 1)//2):
                return prev.data
 
            # Even case
            elif (count % 2 == 0 and
                    currCount == (count//2)+1):
                return (prev.data + current.data)//2
 
            # Update prev for even no. of nodes
            prev = current
 
            #Move to the right
            current = current.right
         
        else:
         
            """ Find the inorder predecessor of current """
            pre = current.left
            while (pre.right != None and
                    pre.right != current):
                pre = pre.right
 
            """ Make current as right child
                of its inorder predecessor """
            if (pre.right == None):
             
                pre.right = current
                current = current.left
            else:
             
                pre.right = None
 
                prev = pre
 
                # Count current node
                currCount+= 1
 
                # Check if the current node is the median
                if (count % 2 != 0 and
                    currCount == (count + 1) // 2 ):
                    return current.data
 
                elif (count%2 == 0 and
                    currCount == (count // 2) + 1):
                    return (prev.data+current.data)//2
 
                # update prev node for the case of even
                # no. of nodes
                prev = current
                current = current.right
 
         
# Driver Code
if __name__ == '__main__':
 
    """ Constructed binary tree is
        50
        / \
    30 70
    / \ / \
    20 40 60 80 """
     
    root = newNode(50)
    insert(root, 30)
    insert(root, 20)
    insert(root, 40)
    insert(root, 70)
    insert(root, 60)
    insert(root, 80)
    print("Median of BST is ",findMedian(root))
 
 
 
# This code is contributed
# Shubham Singh(SHUBHAMSINGH10)

C#




/* C# program to find the median of BST in O(n)
time and O(1) space*/
using System;
class GfG
{
 
/* A binary search tree Node has data, pointer
to left child and a pointer to right child */
public class Node
{
    public int data;
    public Node left, right;
}
 
// A utility function to create a new BST node
static Node newNode(int item)
{
    Node temp = new Node();
    temp.data = item;
    temp.left = null;
    temp.right = null;
    return temp;
}
 
/* A utility function to insert a new node with
given key in BST */
static Node insert(Node node, int key)
{
    /* If the tree is empty, return a new node */
    if (node == null) return newNode(key);
 
    /* Otherwise, recur down the tree */
    if (key < node.data)
        node.left = insert(node.left, key);
    else if (key > node.data)
        node.right = insert(node.right, key);
 
    /* return the (unchanged) node pointer */
    return node;
}
 
/* Function to count nodes in a binary search tree
using Morris Inorder traversal*/
static int counNodes(Node root)
{
    Node current, pre;
 
    // Initialise count of nodes as 0
    int count = 0;
 
    if (root == null)
        return count;
 
    current = root;
    while (current != null)
    {
        if (current.left == null)
        {
            // Count node if its left is NULL
            count++;
 
            // Move to its right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
 
            while (pre.right != null &&
                pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its
            inorder predecessor */
            if(pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if part to
            restore the original tree i.e., fix
            the right child of predecessor */
            else
            {
                pre.right = null;
 
                // Increment count if the current
                // node is to be visited
                count++;
                current = current.right;
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
 
    return count;
}
 
 
/* Function to find median in O(n) time and O(1) space
using Morris Inorder traversal*/
static int findMedian(Node root)
{
if (root == null)
        return 0;
 
    int count = counNodes(root);
    int currCount = 0;
    Node current = root, pre = null, prev = null;
 
    while (current != null)
    {
        if (current.left == null)
        {
            // count current node
            currCount++;
 
            // check if current node is the median
            // Odd case
            if (count % 2 != 0 && currCount == (count+1)/2)
                return prev.data;
 
            // Even case
            else if (count % 2 == 0 && currCount == (count/2)+1)
                return (prev.data + current.data)/2;
 
            // Update prev for even no. of nodes
            prev = current;
 
            //Move to the right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
            while (pre.right != null && pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its inorder predecessor */
            if (pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if part to restore the original
            tree i.e., fix the right child of predecessor */
            else
            {
                pre.right = null;
 
                prev = pre;
 
                // Count current node
                currCount++;
 
                // Check if the current node is the median
                if (count % 2 != 0 && currCount == (count+1)/2 )
                    return current.data;
 
                else if (count % 2 == 0 && currCount == (count/2)+1)
                    return (prev.data + current.data)/2;
 
                // update prev node for the case of even
                // no. of nodes
                prev = current;
                current = current.right;
 
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
    return -1;
}
 
/* Driver code*/
public static void Main(String []args)
{
 
    /* Let us create following BST
                50
            / \
            30 70
            / \ / \
        20 40 60 80 */
    Node root = null;
    root = insert(root, 50);
    insert(root, 30);
    insert(root, 20);
    insert(root, 40);
    insert(root, 70);
    insert(root, 60);
    insert(root, 80);
 
    Console.WriteLine("Median of BST is " + findMedian(root));
}
}
 
// This code is contributed by Arnab Kundu

Javascript




<script>
 
/* JavaScript program to find
the median of BST in O(n)
time and O(1) space*/
 
 
/* A binary search tree Node has data, pointer
to left child and a pointer to right child */
     class Node {
            constructor() {
                this.data = 0;
                this.left = null;
                this.right = null;
            }
        }
 
// A utility function to create a new BST node
function newNode(item)
{
    var temp = new Node();
    temp.data = item;
    temp.left = null;
    temp.right = null;
    return temp;
}
 
/* A utility function to insert a new node with
given key in BST */
function insert(node , key)
{
    /* If the tree is empty, return a new node */
    if (node == null) return newNode(key);
 
    /* Otherwise, recur down the tree */
    if (key < node.data)
        node.left = insert(node.left, key);
    else if (key > node.data)
        node.right = insert(node.right, key);
 
    /* return the (unchanged) node pointer */
    return node;
}
 
/* Function to count nodes in a binary search tree
using Morris Inorder traversal*/
function counNodes(root)
{
    var current, pre;
 
    // Initialise count of nodes as 0
    var count = 0;
 
    if (root == null)
        return count;
 
    current = root;
    while (current != null)
    {
        if (current.left == null)
        {
            // Count node if its left is NULL
            count++;
 
            // Move to its right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
 
            while (pre.right != null &&
                pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its
            inorder predecessor */
            if(pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if part to
            restore the original tree i.e., fix
            the right child of predecessor */
            else
            {
                pre.right = null;
 
                // Increment count if the current
                // node is to be visited
                count++;
                current = current.right;
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
 
    return count;
} /* Function to find median in O(n) time and O(1) space
using Morris Inorder traversal*/
function findMedian(root)
{
if (root == null)
        return 0;
 
    var count = counNodes(root);
    var currCount = 0;
    var current = root, pre = null, prev = null;
 
    while (current != null)
    {
        if (current.left == null)
        {
            // count current node
            currCount++;
 
            // check if current node is the median
            // Odd case
            if (count % 2 != 0 &&
            currCount == (count+1)/2)
                return prev.data;
 
            // Even case
            else if (count % 2 == 0 &&
            currCount == (count/2)+1)
                return (prev.data + current.data)/2;
 
            // Update prev for even no. of nodes
            prev = current;
 
            //Move to the right
            current = current.right;
        }
        else
        {
            /* Find the inorder predecessor of current */
            pre = current.left;
            while (pre.right != null &&
            pre.right != current)
                pre = pre.right;
 
            /* Make current as right child of its
            inorder predecessor */
            if (pre.right == null)
            {
                pre.right = current;
                current = current.left;
            }
 
            /* Revert the changes made in if
            part to restore the original
            tree i.e., fix the right child of predecessor */
            else
            {
                pre.right = null;
 
                prev = pre;
 
                // Count current node
                currCount++;
 
                // Check if the current node is the median
                if (count % 2 != 0 &&
                currCount == (count+1)/2 )
                    return current.data;
 
                else if (count%2==0 &&
                currCount == (count/2)+1)
                    return (prev.data+current.data)/2;
 
                // update prev node for the case of even
                // no. of nodes
                prev = current;
                current = current.right;
 
            } /* End of if condition pre->right == NULL */
        } /* End of if condition current->left == NULL*/
    } /* End of while */
    return -1;
}
 
/* Driver code*/
 
    /* Let us create following BST
                50
            / \
            30 70
            / \ / \
        20 40 60 80 */
    var root = null;
    root = insert(root, 50);
    insert(root, 30);
    insert(root, 20);
    insert(root, 40);
    insert(root, 70);
    insert(root, 60);
    insert(root, 80);
 
    document.write("Median of BST is " + findMedian(root));
 
// This code is contributed by todaysgaurav
 
</script>

Output

Median of BST is 50

Time Complexity: O(N)
Auxiliary Space: O(1)


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