How to implement decrease key or change key in Binary Search Tree?

Given a Binary Search Tree, write a function that takes the following three as arguments: 

1. Root of tree 
2. Old key value 
3. New Key Value 

The function should change old key value to new key value. The function may assume that old key-value always exists in Binary Search Tree.

Example:

Input: Root of below tree
              50
           /     \
          30      70
         /  \    /  \
       20   40  60   80 
     Old key value:  40
     New key value:  10

Output: BST should be modified to following
              50
           /     \
          30      70
         /       /  \
       20      60   80  
       /
     10

We strongly recommend you to minimize your browser and try this yourself first

The idea is to call delete for old key value, then call insert for new key value. Below is C++ implementation of the idea. 

C++

// C++ program to demonstrate decrease
// key operation on binary search tree
#include<bits/stdc++.h>

using namespace std;

class node
{
    public:
    int key;
    node *left, *right;
};

// A utility function to
// create a new BST node
node *newNode(int item)
{
    node *temp = new node;
    temp->key = item;
    temp->left = temp->right = NULL;
    return temp;
}

// A utility function to
// do inorder traversal of BST
void inorder(node *root)
{
    if (root != NULL)
    {
        inorder(root->left);
        cout << root->key << " ";
        inorder(root->right);
    }
}

/* A utility function to insert
a new node with given key in BST */
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->key)
        node->left = insert(node->left, key);
    else
        node->right = insert(node->right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree,
return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
node * minValueNode(node* Node)
{
    node* current = Node;

    /* loop down to find the leftmost leaf */
    while (current->left != NULL)
        current = current->left;

    return current;
}

/* Given a binary search tree and
a key, this function deletes the key
and returns the new root */
node* deleteNode(node* root, int key)
{
    // base case
    if (root == NULL) return root;

    // If the key to be deleted is
    // smaller than the root's key,
    // then it lies in left subtree
    if (key < root->key)
        root->left = deleteNode(root->left, key);

    // If the key to be deleted is
    // greater than the root's key,
    // then it lies in right subtree
    else if (key > root->key)
        root->right = deleteNode(root->right, key);

    // if key is same as root's
    // key, then This is the node
    // to be deleted
    else
    {
        // node with only one child or no child
        if (root->left == NULL)
        {
            node *temp = root->right;
            free(root);
            return temp;
        }
        else if (root->right == NULL)
        {
            node *temp = root->left;
            free(root);
            return temp;
        }

        // node with two children: Get
        // the inorder successor (smallest
        // in the right subtree)
        node* temp = minValueNode(root->right);

        // Copy the inorder successor's
        // content to this node
        root->key = temp->key;

        // Delete the inorder successor
        root->right = deleteNode(root->right, temp->key);
    }
    return root;
}

// Function to decrease a key
// value in Binary Search Tree
node *changeKey(node *root, int oldVal, int newVal)
{
    // First delete old key value
    root = deleteNode(root, oldVal);

    // Then insert new key value
    root = insert(root, newVal);

    // Return new root
    return root;
}

// Driver code
int main()
{
    /* Let us create following BST
            50
        / \
        30 70
        / \ / \
    20 40 60 80 */
    node *root = NULL;
    root = insert(root, 50);
    root = insert(root, 30);
    root = insert(root, 20);
    root = insert(root, 40);
    root = insert(root, 70);
    root = insert(root, 60);
    root = insert(root, 80);


    cout << "Inorder traversal of the given tree \n";
    inorder(root);

    root = changeKey(root, 40, 10);

    /* BST is modified to
            50
        / \
        30 70
        / / \
    20 60 80
    /
    10 */
    cout << "\nInorder traversal of the modified tree \n";
    inorder(root);

    return 0;
}

// This code is contributed by rathbhupendra

C

// C program to demonstrate decrease  key operation on binary search tree
#include<stdio.h>
#include<stdlib.h>

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 do inorder traversal of BST
void inorder(struct node *root)
{
    if (root != NULL)
    {
        inorder(root->left);
        printf("%d ", root->key);
        inorder(root->right);
    }
}

/* 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->key)
        node->left  = insert(node->left, key);
    else
        node->right = insert(node->right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree, return the node with minimum
   key value found in that tree. Note that the entire tree does not
   need to be searched. */
struct node * minValueNode(struct node* node)
{
    struct node* current = node;

    /* loop down to find the leftmost leaf */
    while (current->left != NULL)
        current = current->left;

    return current;
}

/* Given a binary search tree and a key, this function deletes the key
   and returns the new root */
struct node* deleteNode(struct node* root, int key)
{
    // base case
    if (root == NULL) return root;

    // If the key to be deleted is smaller than the root's key,
    // then it lies in left subtree
    if (key < root->key)
        root->left = deleteNode(root->left, key);

    // If the key to be deleted is greater than the root's key,
    // then it lies in right subtree
    else if (key > root->key)
        root->right = deleteNode(root->right, key);

    // if key is same as root's key, then This is the node
    // to be deleted
    else
    {
        // node with only one child or no child
        if (root->left == NULL)
        {
            struct node *temp = root->right;
            free(root);
            return temp;
        }
        else if (root->right == NULL)
        {
            struct node *temp = root->left;
            free(root);
            return temp;
        }

        // node with two children: Get the inorder successor (smallest
        // in the right subtree)
        struct node* temp = minValueNode(root->right);

        // Copy the inorder successor's content to this node
        root->key = temp->key;

        // Delete the inorder successor
        root->right = deleteNode(root->right, temp->key);
    }
    return root;
}

// Function to decrease a key value in Binary Search Tree
struct node *changeKey(struct node *root, int oldVal, int newVal)
{
    //  First delete old key value
    root = deleteNode(root, oldVal);

    // Then insert new key value
    root = insert(root, newVal);

    // Return new root
    return root;
}

// 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);
    root = insert(root, 30);
    root = insert(root, 20);
    root = insert(root, 40);
    root = insert(root, 70);
    root = insert(root, 60);
    root = insert(root, 80);


    printf("Inorder traversal of the given tree \n");
    inorder(root);

    root = changeKey(root, 40, 10);

    /* BST is modified to
              50
           /     \
          30      70
         /       /  \
       20      60   80
       /
     10     */
    printf("\nInorder traversal of the modified tree \n");
    inorder(root);

    return 0;
}

Java

// Java program to demonstrate decrease
// key operation on binary search tree
class GfG
{

static class node
{
    int key;
    node left, right;
}
static node root = null;

// A utility function to
// create a new BST node
static node newNode(int item)
{
    node temp = new node();
    temp.key = item;
    temp.left = null;
    temp.right = null;
    return temp;
}

// A utility function to
// do inorder traversal of BST
static void inorder(node root)
{
    if (root != null)
    {
        inorder(root.left);
        System.out.print(root.key + " ");
        inorder(root.right);
    }
}

/* 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.key)
        node.left = insert(node.left, key);
    else
        node.right = insert(node.right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree,
return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
static node minValueNode(node Node)
{
    node current = Node;

    /* loop down to find the leftmost leaf */
    while (current.left != null)
        current = current.left;

    return current;
}

/* Given a binary search tree and
a key, this function deletes the key
and returns the new root */
static node deleteNode(node root, int key)
{
    // base case
    if (root == null) return root;

    // If the key to be deleted is
    // smaller than the root's key,
    // then it lies in left subtree
    if (key < root.key)
        root.left = deleteNode(root.left, key);

    // If the key to be deleted is
    // greater than the root's key,
    // then it lies in right subtree
    else if (key > root.key)
        root.right = deleteNode(root.right, key);

    // if key is same as root's
    // key, then This is the node
    // to be deleted
    else
    {
        // node with only one child or no child
        if (root.left == null)
        {
            node temp = root.right;
            return temp;
        }
        else if (root.right == null)
        {
            node temp = root.left;
            return temp;
        }

        // node with two children: Get
        // the inorder successor (smallest
        // in the right subtree)
        node temp = minValueNode(root.right);

        // Copy the inorder successor's
        // content to this node
        root.key = temp.key;

        // Delete the inorder successor
        root.right = deleteNode(root.right, temp.key);
    }
    return root;
}

// Function to decrease a key
// value in Binary Search Tree
static node changeKey(node root, int oldVal, int newVal)
{
    // First delete old key value
    root = deleteNode(root, oldVal);

    // Then insert new key value
    root = insert(root, newVal);

    // Return new root
    return root;
}

// Driver code
public static void main(String[] args)
{
    /* Let us create following BST
            50
        / \
        30 70
        / \ / \
    20 40 60 80 */
    root = insert(root, 50);
    root = insert(root, 30);
    root = insert(root, 20);
    root = insert(root, 40);
    root = insert(root, 70);
    root = insert(root, 60);
    root = insert(root, 80);


    System.out.println("Inorder traversal of the given tree");
    inorder(root);

    root = changeKey(root, 40, 10);

    /* BST is modified to
            50
        / \
        30 70
        / / \
    20 60 80
    /
    10 */
    System.out.println("\nInorder traversal of the modified tree ");
    inorder(root);
}
}

// This code is contributed by Prerna saini

Python3

# Python3 program to demonstrate decrease key
# operation on binary search tree

# A utility function to create a new BST node
class newNode:

    def __init__(self, key):
        self.key = key
        self.left = self.right = None

# A utility function to do inorder
# traversal of BST
def inorder(root):
    if root != None:
        inorder(root.left)
        print(root.key,end=" ")
        inorder(root.right)

# 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.key:
        node.left = insert(node.left, key)
    else:
        node.right = insert(node.right, key)

    # return the (unchanged) node pointer
    return node

# Given a non-empty binary search tree, return
# the node with minimum key value found in that
# tree. Note that the entire tree does not
# need to be searched.
def minValueNode(node):
    current = node

    # loop down to find the leftmost leaf
    while current.left != None:
        current = current.left
    return current

# Given a binary search tree and a key, this
# function deletes the key and returns the new root
def deleteNode(root, key):

    # base case
    if root == None:
        return root

    # If the key to be deleted is smaller than
    # the root's key, then it lies in left subtree
    if key < root.key:
        root.left = deleteNode(root.left, key)

    # If the key to be deleted is greater than
    # the root's key, then it lies in right subtree
    elif key > root.key:
        root.right = deleteNode(root.right, key)

    # if key is same as root's key, then
    # this is the node to be deleted
    else:

        # node with only one child or no child
        if root.left == None:
            temp = root.right
            return temp
        elif root.right == None:
            temp = root.left
            return temp

        # node with two children: Get the inorder
        # successor (smallest in the right subtree)
        temp = minValueNode(root.right)

        # Copy the inorder successor's content
        # to this node
        root.key = temp.key

        # Delete the inorder successor
        root.right = deleteNode(root.right, temp.key)
    return root

# Function to decrease a key value in
# Binary Search Tree
def changeKey(root, oldVal, newVal):

    # First delete old key value
    root = deleteNode(root, oldVal)

    # Then insert new key value
    root = insert(root, newVal)

    # Return new root
    return root

# Driver Code
if __name__ == '__main__':

    # Let us create following BST
    #         50
    #     /     \
    #     30     70
    #     / \ / \
    # 20 40 60 80
    root = None
    root = insert(root, 50)
    root = insert(root, 30)
    root = insert(root, 20)
    root = insert(root, 40)
    root = insert(root, 70)
    root = insert(root, 60)
    root = insert(root, 80)

    print("Inorder traversal of the given tree")
    inorder(root)

    root = changeKey(root, 40, 10)
    print()

    # BST is modified to
    #         50
    #     /     \
    #     30     70
    #     /     / \
    # 20     60 80
    # /
    # 10
    print("Inorder traversal of the modified tree")
    inorder(root)

# This code is contributed by PranchalK

C#

// C# program to demonstrate decrease
// key operation on binary search tree
using System;

class GFG
{
public class node
{
    public int key;
    public node left, right;
}
static node root = null;

// A utility function to
// create a new BST node
static node newNode(int item)
{
    node temp = new node();
    temp.key = item;
    temp.left = null;
    temp.right = null;
    return temp;
}

// A utility function to
// do inorder traversal of BST
static void inorder(node root)
{
    if (root != null)
    {
        inorder(root.left);
        Console.Write(root.key + " ");
        inorder(root.right);
    }
}

/* 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.key)
        node.left = insert(node.left, key);
    else
        node.right = insert(node.right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree,
return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
static node minValueNode(node Node)
{
    node current = Node;

    /* loop down to find the leftmost leaf */
    while (current.left != null)
        current = current.left;

    return current;
}

/* Given a binary search tree and
a key, this function deletes the key
and returns the new root */
static node deleteNode(node root, int key)
{
    node temp = null;

    // base case
    if (root == null) return root;

    // If the key to be deleted is
    // smaller than the root's key,
    // then it lies in left subtree
    if (key < root.key)
        root.left = deleteNode(root.left, key);

    // If the key to be deleted is
    // greater than the root's key,
    // then it lies in right subtree
    else if (key > root.key)
        root.right = deleteNode(root.right, key);

    // if key is same as root's
    // key, then This is the node
    // to be deleted
    else
    {

        // node with only one child or no child
        if (root.left == null)
        {
            temp = root.right;
            return temp;
        }
        else if (root.right == null)
        {
            temp = root.left;
            return temp;
        }

        // node with two children: Get
        // the inorder successor (smallest
        // in the right subtree)
        temp = minValueNode(root.right);

        // Copy the inorder successor's
        // content to this node
        root.key = temp.key;

        // Delete the inorder successor
        root.right = deleteNode(root.right,
                                 temp.key);
    }
    return root;
}

// Function to decrease a key
// value in Binary Search Tree
static node changeKey(node root, int oldVal,
                                 int newVal)
{
    // First delete old key value
    root = deleteNode(root, oldVal);

    // Then insert new key value
    root = insert(root, newVal);

    // Return new root
    return root;
}

// Driver code
public static void Main(String[] args)
{
    /* Let us create following BST
            50
        / \
        30 70
        / \ / \
    20 40 60 80 */
    root = insert(root, 50);
    root = insert(root, 30);
    root = insert(root, 20);
    root = insert(root, 40);
    root = insert(root, 70);
    root = insert(root, 60);
    root = insert(root, 80);

    Console.WriteLine("Inorder traversal " +
                      "of the given tree ");
    inorder(root);

    root = changeKey(root, 40, 10);

    /* BST is modified to
            50
        / \
        30 70
        / / \
    20 60 80
    /
    10 */
    Console.WriteLine("\nInorder traversal " +
                      "of the modified tree");
    inorder(root);
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// JavaScript program to demonstrate decrease
// key operation on binary search tree

 class node
{
        constructor() {
            this.key = 0;
            this.left = null;
            this.right = null;
        }
    }
 var root = null;

// A utility function to
// create a new BST node
 function newNode(item)
{
    var temp = new node();
    temp.key = item;
    temp.left = null;
    temp.right = null;
    return temp;
}

// A utility function to
// do inorder traversal of BST
function inorder( root)
{
    if (root != null)
    {
        inorder(root.left);
        document.write(root.key + " ");
        inorder(root.right);
    }
}

/* 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.key)
        node.left = insert(node.left, key);
    else
        node.right = insert(node.right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree,
return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
 function minValueNode( Node)
{
    var current = Node;

    /* loop down to find the leftmost leaf */
    while (current.left != null)
        current = current.left;

    return current;
}

/* Given a binary search tree and
a key, this function deletes the key
and returns the new root */
 function deleteNode( root , key)
{
    // base case
    if (root == null) return root;

    // If the key to be deleted is
    // smaller than the root's key,
    // then it lies in left subtree
    if (key < root.key)
        root.left = deleteNode(root.left, key);

    // If the key to be deleted is
    // greater than the root's key,
    // then it lies in right subtree
    else if (key > root.key)
        root.right = deleteNode(root.right, key);

    // if key is same as root's
    // key, then This is the node
    // to be deleted
    else
    {
        // node with only one child or no child
        if (root.left == null)
        {
             temp = root.right;
            return temp;
        }
        else if (root.right == null)
        {
             temp = root.left;
            return temp;
        }

        // node with two children: Get
        // the inorder successor (smallest
        // in the right subtree)
        var temp = minValueNode(root.right);

        // Copy the inorder successor's
        // content to this node
        root.key = temp.key;

        // Delete the inorder successor
        root.right = deleteNode(root.right, temp.key);
    }
    return root;
}

// Function to decrease a key
// value in Binary Search Tree
 function changeKey( root , oldVal , newVal)
{
    // First delete old key value
    root = deleteNode(root, oldVal);

    // Then insert new key value
    root = insert(root, newVal);

    // Return new root
    return root;
}

// Driver code


    /* Let us create following BST
            50
        / \
        30 70
        / \ / \
    20 40 60 80 */
    root = insert(root, 50);
    root = insert(root, 30);
    root = insert(root, 20);
    root = insert(root, 40);
    root = insert(root, 70);
    root = insert(root, 60);
    root = insert(root, 80);

    document.write(
    "Inorder traversal of the given tree<br/>"
    );
    inorder(root);

    root = changeKey(root, 40, 10);

    /* BST is modified to
            50
        / \
        30 70
        / / \
    20 60 80
    /
    10 */

    document.write(
    "<br/>Inorder traversal of the modified tree <br/>"
    );
    inorder(root);

// This code contributed by aashish1995

</script>

Output

Inorder traversal of the given tree 
20 30 40 50 60 70 80 
Inorder traversal of the modified tree 
10 20 30 50 60 70 80 

Time complexity of above changeKey() is O(h) where h is height of BST. 

Auxiliary Space: O(N),  here N is number of node as we used extra space to store nodes.