K’th Largest Element in BST when modification to BST is not allowed

Given a Binary Search Tree (BST) and a positive integer k, find the k’th largest element in the Binary Search Tree. 
For example, in the following BST, if k = 3, then output should be 14, and if k = 5, then output should be 10. 

We have discussed two methods in this post. The method 1 requires O(n) time. The method 2 takes O(h) time where h is height of BST, but requires augmenting the BST (storing count of nodes in left subtree with every node). 
Can we find k’th largest element in better than O(n) time and no augmentation?

Approach:

1. The idea is to do reverse inorder traversal of BST. Keep a count of nodes visited.
2. The reverse inorder traversal traverses all nodes in decreasing order, i.e, visit the right node then centre then left and continue traversing the nodes recursively.
3. While doing the traversal, keep track of the count of nodes visited so far.
4. When the count becomes equal to k, stop the traversal and print the key.

Implementation:

C++

// C++ program to find k'th largest element in BST
#include<bits/stdc++.h>
using namespace std;

struct Node
{
    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 function to find k'th largest element in a given tree.
void kthLargestUtil(Node *root, int k, int &c)
{
    // Base cases, the second condition is important to
    // avoid unnecessary recursive calls
    if (root == NULL || c >= k)
        return;

    // Follow reverse inorder traversal so that the
    // largest element is visited first
    kthLargestUtil(root->right, k, c);

    // Increment count of visited nodes
    c++;

    // If c becomes k now, then this is the k'th largest
    if (c == k)
    {
        cout << "K'th largest element is "
             << root->key << endl;
        return;
    }

    // Recur for left subtree
    kthLargestUtil(root->left, k, c);
}

// Function to find k'th largest element
void kthLargest(Node *root, int k)
{
    // Initialize count of nodes visited as 0
    int c = 0;

    // Note that c is passed by reference
    kthLargestUtil(root, k, c);
}

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

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

// Driver Program to test above functions
int main()
{
    /* 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);

    int c = 0;
    for (int k=1; k<=7; k++)
        kthLargest(root, k);

    return 0;
}

Java

// Java code to find k'th largest element in BST

// A binary tree node
class Node {

    int data;
    Node left, right;

    Node(int d)
    {
        data = d;
        left = right = null;
    }
}

class BinarySearchTree {

    // Root of BST
    Node root;

    // Constructor
    BinarySearchTree()
    {
        root = null;
    }

    // function to insert nodes
    public void insert(int data)
    {
        this.root = this.insertRec(this.root, data);
    }

    /* A utility function to insert a new node
    with given key in BST */
    Node insertRec(Node node, int data)
    {
        /* If the tree is empty, return a new node */
        if (node == null) {
            this.root = new Node(data);
            return this.root;
        }

        if (data == node.data) {
            return node;
        }

        /* Otherwise, recur down the tree */
        if (data < node.data) {
            node.left = this.insertRec(node.left, data);
        } else {
            node.right = this.insertRec(node.right, data);
        }
        return node;
    }

    // class that stores the value of count
    public class count {
        int c = 0;
    }

    // utility function to find kth largest no in
    // a given tree
    void kthLargestUtil(Node node, int k, count C)
    {
        // Base cases, the second condition is important to
        // avoid unnecessary recursive calls
        if (node == null || C.c >= k)
            return;

        // Follow reverse inorder traversal so that the
        // largest element is visited first
        this.kthLargestUtil(node.right, k, C);

        // Increment count of visited nodes
        C.c++;

        // If c becomes k now, then this is the k'th largest
        if (C.c == k) {
            System.out.println(k + "th largest element is " +
                                                 node.data);
            return;
        }

        // Recur for left subtree
        this.kthLargestUtil(node.left, k, C);
    }

    // Method to find the kth largest no in given BST
    void kthLargest(int k)
    {
        count c = new count(); // object of class count
        this.kthLargestUtil(this.root, k, c);
    }

    // Driver function
    public static void main(String[] args)
    {
        BinarySearchTree tree = new BinarySearchTree();

        /* Let us create following BST
              50
           /     \
          30      70
         /  \    /  \
       20   40  60   80 */
        tree.insert(50);
        tree.insert(30);
        tree.insert(20);
        tree.insert(40);
        tree.insert(70);
        tree.insert(60);
        tree.insert(80);

        for (int i = 1; i <= 7; i++) {
            tree.kthLargest(i);
        }
    }
}

// This code is contributed by Kamal Rawal

Python3

# Python3 program to find k'th largest
# element in BST

class Node:

    # Constructor to create a new node
    def __init__(self, data):
        self.key = data
        self.left = None
        self.right = None

# A function to find k'th largest
# element in a given tree.
def kthLargestUtil(root, k, c):

    # Base cases, the second condition
    # is important to avoid unnecessary
    # recursive calls
    if root == None or c[0] >= k:
        return

    # Follow reverse inorder traversal
    # so that the largest element is
    # visited first
    kthLargestUtil(root.right, k, c)

    # Increment count of visited nodes
    c[0] += 1

    # If c becomes k now, then this is
    # the k'th largest
    if c[0] == k:
        print("K'th largest element is",
                               root.key)
        return

    # Recur for left subtree
    kthLargestUtil(root.left, k, c)

# Function to find k'th largest element
def kthLargest(root, k):

    # Initialize count of nodes
    # visited as 0
    c = [0]

    # Note that c is passed by reference
    kthLargestUtil(root, k, c)

# 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 Node(key)

    # Otherwise, recur down the tree
    if key < node.key:
        node.left = insert(node.left, key)
    elif key > node.key:
        node.right = insert(node.right, key)

    # return the (unchanged) node pointer
    return node

# Driver Code
if __name__ == '__main__':

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

    for k in range(1,8):
        kthLargest(root, k)

# This code is contributed by PranchalK

C#

using System;

// C# code to find k'th largest element in BST

// A binary tree node
public class Node
{

    public int data;
    public Node left, right;

    public Node(int d)
    {
        data = d;
        left = right = null;
    }
}

public class BinarySearchTree
{

    // Root of BST
    public Node root;

    // Constructor
    public BinarySearchTree()
    {
        root = null;
    }

    // function to insert nodes
    public virtual void insert(int data)
    {
        this.root = this.insertRec(this.root, data);
    }

    /* A utility function to insert a new node
    with given key in BST */
    public virtual Node insertRec(Node node, int data)
    {
        /* If the tree is empty, return a new node */
        if (node == null)
        {
            this.root = new Node(data);
            return this.root;
        }

        if (data == node.data)
        {
            return node;
        }

        /* Otherwise, recur down the tree */
        if (data < node.data)
        {
            node.left = this.insertRec(node.left, data);
        }
        else
        {
            node.right = this.insertRec(node.right, data);
        }
        return node;
    }

    // class that stores the value of count
    public class count
    {
        private readonly BinarySearchTree outerInstance;

        public count(BinarySearchTree outerInstance)
        {
            this.outerInstance = outerInstance;
        }

        internal int c = 0;
    }

    // utility function to find kth largest no in
    // a given tree
    public virtual void kthLargestUtil(Node node, int k, count C)
    {
        // Base cases, the second condition is important to
        // avoid unnecessary recursive calls
        if (node == null || C.c >= k)
        {
            return;
        }

        // Follow reverse inorder traversal so that the
        // largest element is visited first
        this.kthLargestUtil(node.right, k, C);

        // Increment count of visited nodes
        C.c++;

        // If c becomes k now, then this is the k'th largest
        if (C.c == k)
        {
            Console.WriteLine(k + "th largest element is " + node.data);
            return;
        }

        // Recur for left subtree
        this.kthLargestUtil(node.left, k, C);
    }

    // Method to find the kth largest no in given BST
    public virtual void kthLargest(int k)
    {
        count c = new count(this); // object of class count
        this.kthLargestUtil(this.root, k, c);
    }

    // Driver function
    public static void Main(string[] args)
    {
        BinarySearchTree tree = new BinarySearchTree();

        /* Let us create following BST
              50
           /     \
          30      70
         /  \    /  \
       20   40  60   80 */
        tree.insert(50);
        tree.insert(30);
        tree.insert(20);
        tree.insert(40);
        tree.insert(70);
        tree.insert(60);
        tree.insert(80);

        for (int i = 1; i <= 7; i++)
        {
            tree.kthLargest(i);
        }
    }
}

  // This code is contributed by Shrikant13

Javascript

<script>
// javascript code to find k'th largest element in BST

// A binary tree node
class Node {

   constructor(d)
    {
        this.data = d;
        this.left = this.right = null;
    }
}



    // Root of BST
    var root = null;

    // Constructor


    // function to insert nodes
    function insert(data)
    {
        this.root = this.insertRec(this.root, data);
    }

    /* A utility function to insert a new node
    with given key in BST */
    function insertRec( node , data)
    {
        /* If the tree is empty, return a new node */
        if (node == null) {
            this.root = new Node(data);
            return this.root;
        }

        if (data == node.data) {
            return node;
        }

        /* Otherwise, recur down the tree */
        if (data < node.data) {
            node.left = this.insertRec(node.left, data);
        } else {
            node.right = this.insertRec(node.right, data);
        }
        return node;
    }

    // class that stores the value of count
     class count {
        constructor(){this.c = 0;}

    }

    // utility function to find kth largest no in
    // a given tree
    function kthLargestUtil( node , k,  C)
    {
        // Base cases, the second condition is important to
        // avoid unnecessary recursive calls
        if (node == null || C.c >= k)
            return;

        // Follow reverse inorder traversal so that the
        // largest element is visited first
        this.kthLargestUtil(node.right, k, C);

        // Increment count of visited nodes
        C.c++;

        // If c becomes k now, then this is the k'th largest
        if (C.c == k) {
            document.write(k + "th largest element is " +
                                                 node.data+"<br/>");
            return;
        }

        // Recur for left subtree
        this.kthLargestUtil(node.left, k, C);
    }

    // Method to find the kth largest no in given BST
    function kthLargest(k)
    {
         c = new count(); // object of class count
        this.kthLargestUtil(this.root, k, c);
    }

    // Driver function



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

        for (i = 1; i <= 7; i++) {
            kthLargest(i);
        }

// This code contributed by gauravrajput1
</script>

Output

K'th largest element is 80
K'th largest element is 70
K'th largest element is 60
K'th largest element is 50
K'th largest element is 40
K'th largest element is 30
K'th largest element is 20

Complexity Analysis:

1. Time Complexity: O(n). 
   In worst case the code can traverse each and every node of the tree if the k given is equal to n (total number of nodes in the tree). Therefore overall time complexity is O(n).
2. Auxiliary Space: O(h). 
   Max recursion stack of height h at a given time.

This article is contributed by Chirag Sharma. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above