Sum of k smallest elements in BST

Given Binary Search Tree. The task is to find sum of all elements smaller than and equal to Kth smallest element.

Examples:

Input :  K = 3
              8
            /   \
           7     10
         /      /   \
        2      9     13
Output : 17
Explanation : Kth smallest element is 8 so sum of all
              element smaller than or equal to 8 are
              2 + 7 + 8

Input : K = 5
           8
         /   \
        5    11
      /  \
     2    7
      \
       3
Output :  25

Method 1 (Does not changes BST node structure): The idea is to traverse BST in inorder traversal. Note that Inorder traversal of BST accesses elements in sorted (or increasing) order. While traversing, we keep track of count of visited Nodes and keep adding Nodes until the count becomes k. 

Implementation:

C++

// c++ program to find Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST
#include <bits/stdc++.h>
using namespace std;

/* Binary tree Node */
struct Node
{
    int data;
    Node* left, * right;
};

// utility function new Node of BST
struct Node *createNode(int data)
{
    Node * new_Node = new Node;
    new_Node->left = NULL;
    new_Node->right = NULL;
    new_Node->data = data;
    return new_Node;
}

// A utility function to insert a new Node
//  with given key in BST and also maintain lcount ,Sum
struct Node * insert(Node *root, int key)
{
    // If the tree is empty, return a new Node
    if (root == NULL)
        return createNode(key);

    // Otherwise, recur down the tree
    if (root->data > key)
        root->left = insert(root->left, key);

    else if (root->data < key)
        root->right = insert(root->right, key);

    // return the (unchanged) Node pointer
    return root;
}

// function return sum of all element smaller than
// and equal to Kth smallest element
int ksmallestElementSumRec(Node *root, int k, int &count)
{
    // Base cases
    if (root == NULL)
        return 0;
    if (count > k)
        return 0;

    // Compute sum of elements in left subtree
    int res = ksmallestElementSumRec(root->left, k, count);
    if (count >= k)
        return res;

    // Add root's data
    res += root->data;

    // Add current Node
    count++;
    if (count >= k)
      return res;

    // If count is less than k, return right subtree Nodes
    return res + ksmallestElementSumRec(root->right, k, count);
}

// Wrapper over ksmallestElementSumRec()
int ksmallestElementSum(struct Node *root, int k)
{
   int count = 0;
   return ksmallestElementSumRec(root, k, count);
}

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

    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node *root = NULL;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    int k = 3;
    cout <<  ksmallestElementSum(root, k) <<endl;
    return 0;
}

Java

// Java program to find Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST
import java.util.*;
class GFG
{

  /* Binary tree Node */
  static class Node
  {
    int data;
    Node left,  right;
  };

  // utility function new Node of BST
  static Node createNode(int data)
  {
    Node  new_Node = new Node();
    new_Node.left = null;
    new_Node.right = null;
    new_Node.data = data;
    return new_Node;
  }

  // A utility function to insert a new Node
  //  with given key in BST and also maintain lcount ,Sum
  static Node  insert(Node root, int key)
  {

    // If the tree is empty, return a new Node
    if (root == null)
      return createNode(key);

    // Otherwise, recur down the tree
    if (root.data > key)
      root.left = insert(root.left, key);
    else if (root.data < key)
      root.right = insert(root.right, key);

    // return the (unchanged) Node pointer
    return root;
  }


  static int count = 0;

  // function return sum of all element smaller than
  // and equal to Kth smallest element
  static int ksmallestElementSumRec(Node root, int k)
  {

    // Base cases
    if (root == null)
      return 0;
    if (count > k)
      return 0;

    // Compute sum of elements in left subtree
    int res = ksmallestElementSumRec(root.left, k);
    if (count >= k)
      return res;

    // Add root's data
    res += root.data;

    // Add current Node
    count++;
    if (count >= k)
      return res;

    // If count is less than k, return right subtree Nodes
    return res + ksmallestElementSumRec(root.right, k);
  }

  // Wrapper over ksmallestElementSumRec()
  static int ksmallestElementSum(Node root, int k)
  {

    int res = ksmallestElementSumRec(root, k);
    return res;
  }

  /* Driver program to test above functions */
  public static void main(String[] args)
  {

    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node root = null;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    int k = 3;
    int count = ksmallestElementSum(root, k);
    System.out.println(count);
  }
}

// This code is contributed by aashish1995

Python3

# Python3 program to find Sum Of All
# Elements smaller than or equal to
# Kth Smallest Element In BST

INT_MAX = 2147483647

# Binary Tree Node
""" utility that allocates a newNode
with the given key """
class createNode:

    # Construct to create a newNode
    def __init__(self, key):
        self.data = key
        self.left = None
        self.right = None

# A utility function to insert a new
# Node with given key in BST and also
# maintain lcount ,Sum
def insert(root, key) :

    # If the tree is empty, return a new Node
    if (root == None) :
        return createNode(key)

    # Otherwise, recur down the tree
    if (root.data > key) :
        root.left = insert(root.left, key)

    else if (root.data < key):
        root.right = insert(root.right, key)

    # return the (unchanged) Node pointer
    return root

# function return sum of all element smaller
# than and equal to Kth smallest element
def ksmallestElementSumRec(root, k, count) :

    # Base cases
    if (root == None) :
        return 0
    if (count[0] > k[0]) :
        return 0

    # Compute sum of elements in left subtree
    res = ksmallestElementSumRec(root.left, k, count)
    if (count[0] >= k[0]) :
        return res

    # Add root's data
    res += root.data

    # Add current Node
    count[0] += 1
    if (count[0] >= k[0]) :
        return res

    # If count is less than k, return
    # right subtree Nodes
    return res + ksmallestElementSumRec(root.right,
                                        k, count)

# Wrapper over ksmallestElementSumRec()
def ksmallestElementSum(root, k):
    count = [0]
    return ksmallestElementSumRec(root, k, count)

# Driver Code
if __name__ == '__main__':

    """ 20
        / \
    8 22
    / \
    4 12
        / \
        10 14
        """
    root = None
    root = insert(root, 20)
    root = insert(root, 8)
    root = insert(root, 4)
    root = insert(root, 12)
    root = insert(root, 10)
    root = insert(root, 14)
    root = insert(root, 22)

    k = [3]
    print(ksmallestElementSum(root, k))

# This code is contributed by
# Shubham Singh(SHUBHAMSINGH10)

C#

// C# program to find Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST
using System;

public class GFG
{

  /* Binary tree Node */
  public class Node
  {
    public int data;
    public Node left,  right;
  };

  // utility function new Node of BST
  static Node createNode(int data)
  {
    Node  new_Node = new Node();
    new_Node.left = null;
    new_Node.right = null;
    new_Node.data = data;
    return new_Node;
  }

  // A utility function to insert a new Node
  //  with given key in BST and also maintain lcount ,Sum
  static Node  insert(Node root, int key)
  {

    // If the tree is empty, return a new Node
    if (root == null)
      return createNode(key);

    // Otherwise, recur down the tree
    if (root.data > key)
      root.left = insert(root.left, key);
    else if (root.data < key)
      root.right = insert(root.right, key);

    // return the (unchanged) Node pointer
    return root;
  }
  static int count = 0;

  // function return sum of all element smaller than
  // and equal to Kth smallest element
  static int ksmallestElementSumRec(Node root, int k)
  {

    // Base cases
    if (root == null)
      return 0;
    if (count > k)
      return 0;

    // Compute sum of elements in left subtree
    int res = ksmallestElementSumRec(root.left, k);
    if (count >= k)
      return res;

    // Add root's data
    res += root.data;

    // Add current Node
    count++;
    if (count >= k)
      return res;

    // If count is less than k, return right subtree Nodes
    return res + ksmallestElementSumRec(root.right, k);
  }

  // Wrapper over ksmallestElementSumRec()
  static int ksmallestElementSum(Node root, int k)
  {

    int res = ksmallestElementSumRec(root, k);
    return res;
  }

  /* Driver program to test above functions */
  public static void Main(String[] args)
  {

    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node root = null;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);
    int k = 3;
    int count = ksmallestElementSum(root, k);
    Console.WriteLine(count);
  }
}

// This code is contributed by aashish1995

Javascript

<script>

// JavaScript program to find
// Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST
 /* Binary tree Node */
class Node {
    constructor() {
        this.data = 0;
        this.left = null;
        this.right = null;
    }
}

  // utility function new Node of BST
  function createNode(data)
  {
    var  new_Node = new Node();
    new_Node.left = null;
    new_Node.right = null;
    new_Node.data = data;
    return new_Node;
  }

  // A utility function to insert a new Node
  //  with given key in BST and also maintain lcount ,Sum
  function  insert(root , key)
  {

    // If the tree is empty, return a new Node
    if (root == null)
      return createNode(key);

    // Otherwise, recur down the tree
    if (root.data > key)
      root.left = insert(root.left, key);
    else if (root.data < key)
      root.right = insert(root.right, key);

    // return the (unchanged) Node pointer
    return root;
  }
  var count = 0;

  // function return sum of all element smaller than
  // and equal to Kth smallest element
  function ksmallestElementSumRec(root , k)
  {

    // Base cases
    if (root == null)
      return 0;
    if (count > k)
      return 0;

    // Compute sum of elements in left subtree
    var res = ksmallestElementSumRec(root.left, k);
    if (count >= k)
      return res;

    // Add root's data
    res += root.data;

    // Add current Node
    count++;
    if (count >= k)
      return res;

    // If count is less than k, return right subtree Nodes
    return res + ksmallestElementSumRec(root.right, k);
  }

  // Wrapper over ksmallestElementSumRec()
  function ksmallestElementSum(root , k)
  {

    var res = ksmallestElementSumRec(root, k);
    return res;
  }

  /* Driver program to test above functions */



    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    var root = null;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    var k = 3;
    var count = ksmallestElementSum(root, k);
    document.write(count);

// This code is contributed by todaysgaurav

</script>

Output

22

Time complexity: O(k)
Auxiliary Space: O(h), where h is the height of the tree

Method 2 (Efficient and changes structure of BST):

We can find the required sum in O(h) time where h is height of BST. Idea is similar to Kth-th smallest element in BST . Here we use augmented tree data structure to solve this problem efficiently in O(h) time [ h is height of BST ] .

Algorithm : 

BST Node contain to extra fields : Lcount , Sum

For each Node of BST
lCount : store how many left child it has
Sum     : store sum of all left child it has

Find Kth smallest element
[ temp_sum store sum of all element less than equal to K ]

ksmallestElementSumRec(root, K, temp_sum)

  IF root -> lCount == K + 1
      temp_sum += root->data + root->sum;
      break;
  ELSE
     IF k > root->lCount   // Goto right sub-tree
        temp_sum += root->data + root-> sum;
        ksmallestElementSumRec(root->right, K-root->lcount+1, temp_sum)
     ELSE
        // Goto left sub-tree
        ksmallestElementSumRec( root->left, K, temp_sum)

Below is implementation of above algo : 

C++

// C++ program to find Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST
#include <bits/stdc++.h>
using namespace std;

/* Binary tree Node */
struct Node
{
    int data;
    int lCount;
    int Sum ;
    Node* left;
    Node* right;
};

//utility function new Node of BST
struct Node *createNode(int data)
{
    Node * new_Node = new Node;
    new_Node->left = NULL;
    new_Node->right = NULL;
    new_Node->data = data;
    new_Node->lCount = 0 ;
    new_Node->Sum = 0;
    return new_Node;
}

// A utility function to insert a new Node with
// given key in BST and also maintain lcount ,Sum
struct Node * insert(Node *root, int key)
{
    // If the tree is empty, return a new Node
    if (root == NULL)
        return createNode(key);

    // Otherwise, recur down the tree
    if (root->data > key)
    {
        // increment lCount of current Node
        root->lCount++;

        // increment current Node sum by adding
        // key into it
        root->Sum += key;

        root->left= insert(root->left , key);
    }
    else if (root->data < key )
        root->right= insert (root->right , key );

    // return the (unchanged) Node pointer
    return root;
}

// function return sum of all element smaller than and equal
// to Kth smallest element
void ksmallestElementSumRec(Node *root, int k , int &temp_sum)
{
    if (root == NULL)
        return ;

    // if we fount k smallest element then break the function
    if ((root->lCount + 1) == k)
    {
        temp_sum += root->data + root->Sum ;
        return ;
    }

    else if (k > root->lCount)
    {
        // store sum of all element smaller than current root ;
        temp_sum += root->data + root->Sum;

        // decremented k and call right sub-tree
        k = k -( root->lCount + 1);
        ksmallestElementSumRec(root->right , k , temp_sum);
    }
    else // call left sub-tree
        ksmallestElementSumRec(root->left , k , temp_sum );
}

// Wrapper over ksmallestElementSumRec()
int ksmallestElementSum(struct Node *root, int k)
{
    int sum = 0;
    ksmallestElementSumRec(root, k, sum);
    return sum;
}

/* Driver program to test above functions */
int main()
{
    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node *root = NULL;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    int k = 3;
    cout <<  ksmallestElementSum(root, k) << endl;
    return 0;
}

Java

// Java program to find Sum Of All Elements smaller
// than or equal to Kth Smallest Element In BST

import java.util.*;

class GFG{

/* Binary tree Node */
static class Node
{
    int data;
    int lCount;
    int Sum ;
    Node left;
    Node right;
};

//utility function new Node of BST
static Node createNode(int data)
{
    Node  new_Node = new Node();
    new_Node.left = null;
    new_Node.right = null;
    new_Node.data = data;
    new_Node.lCount = 0 ;
    new_Node.Sum = 0;
    return new_Node;
}

// A utility function to insert a new Node with
// given key in BST and also maintain lcount ,Sum
static Node  insert(Node root, int key)
{
    // If the tree is empty, return a new Node
    if (root == null)
        return createNode(key);

    // Otherwise, recur down the tree
    if (root.data > key)
    {
        // increment lCount of current Node
        root.lCount++;

        // increment current Node sum by adding
        // key into it
        root.Sum += key;

        root.left= insert(root.left , key);
    }
    else if (root.data < key )
        root.right= insert (root.right , key );

    // return the (unchanged) Node pointer
    return root;
}

static int temp_sum;
// function return sum of all element smaller than and equal
// to Kth smallest element
static void ksmallestElementSumRec(Node root, int k )
{
    if (root == null)
        return ;

    // if we fount k smallest element then break the function
    if ((root.lCount + 1) == k)
    {
        temp_sum += root.data + root.Sum ;
        return ;
    }

    else if (k > root.lCount)
    {
        // store sum of all element smaller than current root ;
        temp_sum += root.data + root.Sum;

        // decremented k and call right sub-tree
        k = k -( root.lCount + 1);
        ksmallestElementSumRec(root.right , k );
    }
    else // call left sub-tree
        ksmallestElementSumRec(root.left , k );
}

// Wrapper over ksmallestElementSumRec()
static void ksmallestElementSum(Node root, int k)
{
    temp_sum = 0;
    ksmallestElementSumRec(root, k);
}

/* Driver program to test above functions */
public static void main(String[] args)
{
    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node root = null;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    int k = 3;
     ksmallestElementSum(root, k);
     System.out.println(temp_sum);
}
}

// This code is contributed by gauravrajput1

Python3

# Python3 program to find Sum Of All Elements
# smaller than or equal to Kth Smallest Element In BST

# utility function new Node of BST
class createNode:

    # Constructor to create a new node
    def __init__(self, data):
        self.data = data
        self.lCount = 0
        self.Sum = 0
        self.left = None
        self.right = None

# A utility function to insert a new Node with
# given key in BST and also maintain lcount ,Sum
def insert(root, key):

    # If the tree is empty, return a new Node
    if root == None:
        return createNode(key)

    # Otherwise, recur down the tree
    if root.data > key:

        # increment lCount of current Node
        root.lCount += 1

        # increment current Node sum by
        # adding key into it
        root.Sum += key

        root.left= insert(root.left , key)
    else if root.data < key:
        root.right= insert (root.right , key)

    # return the (unchanged) Node pointer
    return root

# function return sum of all element smaller
# than and equal to Kth smallest element
def ksmallestElementSumRec(root, k , temp_sum):
    if root == None:
        return

    # if we fount k smallest element
    # then break the function
    if (root.lCount + 1) == k:
        temp_sum[0] += root.data + root.Sum
        return

    else if k > root.lCount:

        # store sum of all element smaller
        # than current root ;
        temp_sum[0] += root.data + root.Sum

        # decremented k and call right sub-tree
        k = k -( root.lCount + 1)
        ksmallestElementSumRec(root.right,
                               k, temp_sum)
    else: # call left sub-tree
        ksmallestElementSumRec(root.left,
                               k, temp_sum)

# Wrapper over ksmallestElementSumRec()
def ksmallestElementSum(root, k):
    Sum = [0]
    ksmallestElementSumRec(root, k, Sum)
    return Sum[0]

# Driver Code
if __name__ == '__main__':

    # 20
    # / \
    # 8     22
    # / \
    #4     12
    #     / \
    # 10 14
    #
    root = None
    root = insert(root, 20)
    root = insert(root, 8)
    root = insert(root, 4)
    root = insert(root, 12)
    root = insert(root, 10)
    root = insert(root, 14)
    root = insert(root, 22)

    k = 3
    print(ksmallestElementSum(root, k))

# This code is contributed by PranchalK

C#

// C# program to find Sum Of All Elements smaller
// than or equal tp Kth Smallest Element In BST
using System;
public class GFG
{

/* Binary tree Node */
public class Node
{
   public int data;
   public int lCount;
   public int Sum ;
   public Node left;
   public Node right;
};

// utility function new Node of BST
static Node createNode(int data)
{
    Node  new_Node = new Node();
    new_Node.left = null;
    new_Node.right = null;
    new_Node.data = data;
    new_Node.lCount = 0 ;
    new_Node.Sum = 0;
    return new_Node;
}

// A utility function to insert a new Node with
// given key in BST and also maintain lcount ,Sum
static Node  insert(Node root, int key)
{

    // If the tree is empty, return a new Node
    if (root == null)
        return createNode(key);

    // Otherwise, recur down the tree
    if (root.data > key)
    {

        // increment lCount of current Node
        root.lCount++;

        // increment current Node sum by adding
        // key into it
        root.Sum += key;
        root.left = insert(root.left , key);
    }
    else if (root.data < key )
        root.right = insert (root.right , key );

    // return the (unchanged) Node pointer
    return root;
}

static int temp_sum;

// function return sum of all element smaller than and equal
// to Kth smallest element
static void ksmallestElementSumRec(Node root, int k )
{
    if (root == null)
        return ;

    // if we fount k smallest element then break the function
    if ((root.lCount + 1) == k)
    {
        temp_sum += root.data + root.Sum ;
        return ;
    }

    else if (k > root.lCount)
    {

        // store sum of all element smaller than current root ;
        temp_sum += root.data + root.Sum;

        // decremented k and call right sub-tree
        k = k -( root.lCount + 1);
        ksmallestElementSumRec(root.right , k );
    }
    else // call left sub-tree
        ksmallestElementSumRec(root.left , k );
}

// Wrapper over ksmallestElementSumRec()
static void ksmallestElementSum(Node root, int k)
{
    temp_sum = 0;
    ksmallestElementSumRec(root, k);
}

/* Driver program to test above functions */
public static void Main(String[] args)
{
    /*    20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
    Node root = null;
    root = insert(root, 20);
    root = insert(root, 8);
    root = insert(root, 4);
    root = insert(root, 12);
    root = insert(root, 10);
    root = insert(root, 14);
    root = insert(root, 22);

    int k = 3;
     ksmallestElementSum(root, k);
     Console.WriteLine(temp_sum);
}
}

// This code is contributed by gauravrajput1

Javascript

<script>

      // JavaScript program to find Sum Of All Elements smaller
      // than or equal to Kth Smallest Element In BST
      /* Binary tree Node */
      class Node {
        constructor() {
          this.data = 0;
          this.lCount = 0;
          this.Sum = 0;
          this.left = null;
          this.right = null;
        }
      }

      // utility function new Node of BST
      function createNode(data) {
        var new_Node = new Node();
        new_Node.left = null;
        new_Node.right = null;
        new_Node.data = data;
        new_Node.lCount = 0;
        new_Node.Sum = 0;
        return new_Node;
      }

      // A utility function to insert a new Node with
      // given key in BST and also maintain lcount ,Sum
      function insert(root, key) {
        // If the tree is empty, return a new Node
        if (root == null) return createNode(key);

        // Otherwise, recur down the tree
        if (root.data > key) {
          // increment lCount of current Node
          root.lCount++;

          // increment current Node sum by adding
          // key into it
          root.Sum += key;
          root.left = insert(root.left, key);
        } else if (root.data < key)
        root.right = insert(root.right, key);

        // return the (unchanged) Node pointer
        return root;
      }

      var temp_sum = 0;

      // function return sum of all element smaller than and equal
      // to Kth smallest element
      function ksmallestElementSumRec(root, k) {
        if (root == null) return;

        // if we fount k smallest element then break the function
        if (root.lCount + 1 == k) {
          temp_sum += root.data + root.Sum;
          return;
        } else if (k > root.lCount) {
          // store sum of all element smaller than current root ;
          temp_sum += root.data + root.Sum;

          // decremented k and call right sub-tree
          k = k - (root.lCount + 1);
          ksmallestElementSumRec(root.right, k);
        } // call left sub-tree
        else ksmallestElementSumRec(root.left, k);
      }

      // Wrapper over ksmallestElementSumRec()
      function ksmallestElementSum(root, k) {
        temp_sum = 0;
        ksmallestElementSumRec(root, k);
      }

      /* Driver program to test above functions */
      /*  20
        /    \
       8     22
     /   \
    4     12
         /   \
        10    14
          */
      var root = null;
      root = insert(root, 20);
      root = insert(root, 8);
      root = insert(root, 4);
      root = insert(root, 12);
      root = insert(root, 10);
      root = insert(root, 14);
      root = insert(root, 22);

      var k = 3;
      ksmallestElementSum(root, k);
      document.write(temp_sum);

</script>

Output

22

Time Complexity: O(h) where h is height of tree. 

Auxiliary Space: O(1)

Playlist : Trees | Data Structures & Algorithms | Programming Tutorials | GeeksforGeeks

  If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.