# 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 then or equal to 8 are
2 + 7 + 8

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

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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.

## C++

 `// c++ program to find Sum Of All Elements smaller ` `// than or equal to Kth Smallest Element In BST ` `#include ` `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; ` `   ``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) <

## 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)  ` ` `  `    ``elif` `(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) `

Output :

`22`

Time complexity : O(k)

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 sun-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 t Kth Smallest Element In BST ` `#include ` `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; ` `} `

## Python3

 `# Python3 program to find Sum Of All Elements  ` `# smaller than or equal t 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) ` `    ``elif` `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` ` `  `    ``elif` `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 `

Output:

```22
```

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