# Find k-th smallest element in BST (Order Statistics in BST)

• Difficulty Level : Medium
• Last Updated : 16 Jun, 2022

Given the root of a binary search tree and K as input, find Kth smallest element in BST.
For example, in the following BST, if k = 3, then the output should be 10, and if k = 5, then the output should be 14. Method 1: Using Inorder Traversal (O(n) time and O(h) auxiliary space)

The Inorder Traversal of a BST traverses the nodes in increasing order. So the idea is to traverse the tree in Inorder. While traversing, keep track of the count of the nodes visited. If the count becomes k, print the node.

## C++

 `// A simple inorder traversal based C++ program to find k-th``// smallest element in a BST.``#include ` `using` `namespace` `std;` `// A BST node``struct` `Node {``    ``int` `data;``    ``Node *left, *right;``    ``Node(``int` `x)``    ``{``        ``data = x;``        ``left = right = NULL;``    ``}``};` `// Recursive function to insert an key into BST``Node* insert(Node* root, ``int` `x)``{``    ``if` `(root == NULL)``        ``return` `new` `Node(x);``    ``if` `(x < root->data)``        ``root->left = insert(root->left, x);``    ``else` `if` `(x > root->data)``        ``root->right = insert(root->right, x);``    ``return` `root;``}` `// Function to find k'th smallest element in BST``// Here count denotes the number of nodes processed so far``int` `count = 0;``Node* kthSmallest(Node* root, ``int``& k)``{``    ``// base case``    ``if` `(root == NULL)``        ``return` `NULL;` `    ``// search in left subtree``    ``Node* left = kthSmallest(root->left, k);` `    ``// if k'th smallest is found in left subtree, return it``    ``if` `(left != NULL)``        ``return` `left;` `    ``// if current element is k'th smallest, return it``    ``count++;``    ``if` `(count == k)``        ``return` `root;` `    ``// else search in right subtree``    ``return` `kthSmallest(root->right, k);``}` `// Function to print k'th smallest element in BST``void` `printKthSmallest(Node* root, ``int` `k)``{``    ``// maintain index to count number of nodes processed so far` `    ``Node* res = kthSmallest(root, k);``    ``if` `(res == NULL)``        ``cout << ``"There are less than k nodes in the BST"``;``    ``else``        ``cout << ``"K-th Smallest Element is "` `<< res->data;``}` `// main function``int` `main()``{``    ``Node* root = NULL;``    ``int` `keys[] = { 20, 8, 22, 4, 12, 10, 14 };` `    ``for` `(``int` `x : keys)``        ``root = insert(root, x);` `    ``int` `k = 3;``    ``printKthSmallest(root, k);``    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## C

 `// A simple inorder traversal based C++ program to find k-th``// smallest element in a BST.``#include ``#include ` `// A BST node``typedef` `struct` `Node {``    ``int` `data;``    ``struct` `Node *left, *right;``} Node;` `struct` `Node* new_node(``int` `x)``{``    ``struct` `Node* p = ``malloc``(``sizeof``(``struct` `Node));``    ``p->data = x;``    ``p->left = NULL;``    ``p->right = NULL;``    ``return` `p;``}` `// Recursive function to insert an key into BST``Node* insert(Node* root, ``int` `x)``{``    ``if` `(root == NULL)``        ``return` `new_node(x);``    ``if` `(x < root->data)``        ``root->left = insert(root->left, x);``    ``else` `if` `(x > root->data)``        ``root->right = insert(root->right, x);``    ``return` `root;``}` `// Function to find k'th smallest element in BST``// Here count denotes the number of nodes processed so far``int` `count = 0;``Node* kthSmallest(Node* root, ``int` `k)``{``    ``// base case``    ``if` `(root == NULL)``        ``return` `NULL;` `    ``// search in left subtree``    ``Node* left = kthSmallest(root->left, k);` `    ``// if k'th smallest is found in left subtree, return it``    ``if` `(left != NULL)``        ``return` `left;` `    ``// if current element is k'th smallest, return it``    ``count++;``    ``if` `(count == k)``        ``return` `root;` `    ``// else search in right subtree``    ``return` `kthSmallest(root->right, k);``}` `// Function to print k'th smallest element in BST``void` `printKthSmallest(Node* root, ``int` `k)``{``    ``// maintain index to count number of nodes processed so far``    ``Node* res = kthSmallest(root, k);``    ``if` `(res == NULL)``        ``printf``(``"There are less than k nodes in the BST"``);``    ``else``        ``printf``(``"K-th Smallest Element is %d"``, res->data);``}` `// main function``int` `main()``{``    ``Node* root = NULL;``    ``int` `keys[] = { 20, 8, 22, 4, 12, 10, 14 };``    ``int` `keys_size = ``sizeof``(keys) / ``sizeof``(keys);` `    ``for` `(``int` `i = 0; i < keys_size; i++)``        ``root = insert(root, keys[i]);` `    ``int` `k = 3;``    ``printKthSmallest(root, k);``    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Java

 `// A simple inorder traversal based Java program``// to find k-th smallest element in a BST.` `import` `java.io.*;``// A BST node``class` `Node {``    ``int` `data;``    ``Node left, right;``    ``Node(``int` `x)``    ``{``        ``data = x;``        ``left = right = ``null``;``    ``}``}` `class` `GFG {` `    ``static` `int` `count = ``0``;``    ``// Recursive function to insert an key into BST``    ``public` `static` `Node insert(Node root, ``int` `x)``    ``{``        ``if` `(root == ``null``)``            ``return` `new` `Node(x);``        ``if` `(x < root.data)``            ``root.left = insert(root.left, x);``        ``else` `if` `(x > root.data)``            ``root.right = insert(root.right, x);``        ``return` `root;``    ``}` `    ``// Function to find k'th largest element in BST``    ``// Here count denotes the number of nodes processed so far``    ``public` `static` `Node kthSmallest(Node root, ``int` `k)``    ``{``        ``// base case``        ``if` `(root == ``null``)``            ``return` `null``;` `        ``// search in left subtree``        ``Node left = kthSmallest(root.left, k);` `        ``// if k'th smallest is found in left subtree, return it``        ``if` `(left != ``null``)``            ``return` `left;` `        ``// if current element is k'th smallest, return it``        ``count++;``        ``if` `(count == k)``            ``return` `root;` `        ``// else search in right subtree``        ``return` `kthSmallest(root.right, k);``    ``}` `    ``// Function to find k'th largest element in BST``    ``public` `static` `void` `printKthSmallest(Node root, ``int` `k)``    ``{``        ``Node res = kthSmallest(root, k);``        ``if` `(res == ``null``)``            ``System.out.println(``"There are less than k nodes in the BST"``);``        ``else``            ``System.out.println(``"K-th Smallest Element is "` `+ res.data);``    ``}` `    ``public` `static` `void` `main(String[] args)``    ``{``        ``Node root = ``null``;``        ``int` `keys[] = { ``20``, ``8``, ``22``, ``4``, ``12``, ``10``, ``14` `};``        ``for` `(``int` `x : keys)``            ``root = insert(root, x);``        ``int` `k = ``3``;``        ``printKthSmallest(root, k);``    ``}``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Python3

 `# A simple inorder traversal based Python3``# program to find k-th smallest element``# in a BST.` `# A BST node``class` `Node:``    ` `    ``def` `__init__(``self``, key):``        ` `        ``self``.data ``=` `key``        ``self``.left ``=` `None``        ``self``.right ``=` `None` `# Recursive function to insert an key into BST``def` `insert(root, x):``    ` `    ``if` `(root ``=``=` `None``):``        ``return` `Node(x)``    ``if` `(x < root.data):``        ``root.left ``=` `insert(root.left, x)``    ``elif` `(x > root.data):``        ``root.right ``=` `insert(root.right, x)``    ``return` `root` `# Function to find k'th largest element``# in BST. Here count denotes the number``# of nodes processed so far``def` `kthSmallest(root):``    ` `    ``global` `k``    ` `    ``# Base case``    ``if` `(root ``=``=` `None``):``        ``return` `None` `    ``# Search in left subtree``    ``left ``=` `kthSmallest(root.left)` `    ``# If k'th smallest is found in``    ``# left subtree, return it``    ``if` `(left !``=` `None``):``        ``return` `left``        ` `    ``# If current element is k'th``    ``# smallest, return it``    ``k ``-``=` `1``    ``if` `(k ``=``=` `0``):``        ``return` `root` `    ``# Else search in right subtree``    ``return` `kthSmallest(root.right)` `# Function to find k'th largest element in BST``def` `printKthSmallest(root):``    ` `    ``# Maintain index to count number``    ``# of nodes processed so far``    ``count ``=` `0``    ``res ``=` `kthSmallest(root)``    ` `    ``if` `(res ``=``=` `None``):``        ``print``(``"There are less than k nodes in the BST"``)``    ``else``:``        ``print``(``"K-th Smallest Element is "``, res.data)` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``root ``=` `None``    ``keys ``=` `[ ``20``, ``8``, ``22``, ``4``, ``12``, ``10``, ``14` `]` `    ``for` `x ``in` `keys:``        ``root ``=` `insert(root, x)` `    ``k ``=` `3``    ` `    ``printKthSmallest(root)` `# This code is contributed by mohit kumar 29`

## C#

 `// A simple inorder traversal``// based C# program to find``// k-th smallest element in a BST.``using` `System;` `// A BST node``class` `Node{` `public` `int` `data;``public` `Node left, right;``public` `Node(``int` `x)``{``  ``data = x;``  ``left = right = ``null``;``}``}` `class` `GFG{``   ` `static` `int` `count = 0;` `// Recursive function to ``// insert an key into BST``public` `static` `Node insert(Node root,``                          ``int` `x)``{``  ``if` `(root == ``null``)``    ``return` `new` `Node(x);``  ``if` `(x < root.data)``    ``root.left = insert(root.left, x);``  ``else` `if` `(x > root.data)``    ``root.right = insert(root.right, x);``  ``return` `root;``}``     ` `// Function to find k'th largest``// element in BST. Here count``// denotes the number of nodes``// processed so far``public` `static` `Node kthSmallest(Node root,``                               ``int` `k)``{``  ``// base case``  ``if` `(root == ``null``)``    ``return` `null``;` `  ``// search in left subtree``  ``Node left = kthSmallest(root.left, k);` `  ``// if k'th smallest is found``  ``// in left subtree, return it``  ``if` `(left != ``null``)``    ``return` `left;` `  ``// if current element is``  ``// k'th smallest, return it``  ``count++;``  ``if` `(count == k)``    ``return` `root;` `  ``// else search in right subtree``  ``return` `kthSmallest(root.right, k);``}` `// Function to find k'th largest``// element in BST``public` `static` `void` `printKthSmallest(Node root,``                                    ``int` `k)``{``  ``// Maintain an index to``  ``// count number of nodes``  ``// processed so far``  ``count = 0;` `  ``Node res = kthSmallest(root, k);``  ` `  ``if` `(res == ``null``)``    ``Console.WriteLine(``"There are less "` `+``                      ``"than k nodes in the BST"``);``  ``else``    ``Console.WriteLine(``"K-th Smallest"` `+``                      ``" Element is "` `+ res.data);``}` `// Driver code``public` `static` `void` `Main(String[] args)``{` `  ``Node root = ``null``;``  ``int` `[]keys = {20, 8, 22, 4,``                ``12, 10, 14};``  ` `  ``foreach` `(``int` `x ``in` `keys)``    ``root = insert(root, x);` `  ``int` `k = 3;``  ``printKthSmallest(root, k);``}``}` `// This code is contributed by gauravrajput1`

## Javascript

 ``

Output:

`K-th Smallest Element is 10`

We can optimize space using Morris Traversal. Please refer K’th smallest element in BST using O(1) Extra Space for details.

Method 2: Augmented Tree Data Structure (O(h) Time Complexity and O(h) auxiliary space)

The idea is to maintain the rank of each node. We can keep track of elements in the left subtree of every node while building the tree. Since we need the K-th smallest element, we can maintain the number of elements of the left subtree in every node.
Assume that the root is having ‘lCount’ nodes in its left subtree. If K = lCount + 1, root is K-th node. If K < lCount + 1, we will continue our search (recursion) for the Kth smallest element in the left subtree of root. If K > lCount + 1, we continue our search in the right subtree for the (K – lCount – 1)-th smallest element. Note that we need the count of elements in the left subtree only.

## C++

 `// A simple inorder traversal based C++ program to find k-th``// smallest element in a BST.``#include ``using` `namespace` `std;` `// A BST node``struct` `Node {``    ``int` `data;``    ``Node *left, *right;``    ``int` `lCount;``    ``Node(``int` `x)``    ``{``        ``data = x;``        ``left = right = NULL;``        ``lCount = 0;``    ``}``};` `// Recursive function to insert an key into BST``Node* insert(Node* root, ``int` `x)``{``    ``if` `(root == NULL)``        ``return` `new` `Node(x);` `    ``// If a node is inserted in left subtree, then lCount of``    ``// this node is increased. For simplicity, we are``    ``// assuming that all keys (tried to be inserted) are``    ``// distinct.``    ``if` `(x < root->data) {``        ``root->left = insert(root->left, x);``        ``root->lCount++;``    ``}` `    ``else` `if` `(x > root->data)``        ``root->right = insert(root->right, x);``    ``return` `root;``}` `// Function to find k'th smallest element in BST``// Here count denotes the number of nodes processed so far``Node* kthSmallest(Node* root, ``int` `k)``{``    ``// base case``    ``if` `(root == NULL)``        ``return` `NULL;``    ``int` `count = root->lCount + 1;``    ``if` `(count == k)``        ``return` `root;``    ``if` `(count > k)``        ``return` `kthSmallest(root->left, k);``    ``// else search in right subtree``    ``return` `kthSmallest(root->right, k - count);``}` `// main function``int` `main()``{``    ``Node* root = NULL;``    ``int` `keys[] = { 20, 8, 22, 4, 12, 10, 14 };``    ``for` `(``int` `x : keys)``        ``root = insert(root, x);``    ``int` `k = 4;``    ``Node* res = kthSmallest(root, k);``    ``if` `(res == NULL)``        ``cout << ``"There are less than k nodes in the BST"``;``    ``else``        ``cout << ``"K-th Smallest Element is "` `<< res->data;``    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## C

 `// A simple inorder traversal based C++ program to find k-th``// smallest element in a BST.``#include ``#include ` `// A BST node``typedef` `struct` `Node {``    ``int` `data;``    ``struct` `Node *left, *right;``    ``int` `lCount;``} Node;` `Node* new_node(``int` `x)``{``    ``Node* newNode = ``malloc``(``sizeof``(Node));``    ``newNode->data = x;``    ``newNode->left = NULL;``    ``newNode->right = NULL;``    ``return` `newNode;``}` `// Recursive function to insert an key into BST``Node* insert(Node* root, ``int` `x)``{``    ``if` `(root == NULL)``        ``return` `new_node(x);` `    ``// If a node is inserted in left subtree, then lCount of``    ``// this node is increased. For simplicity, we are``    ``// assuming that all keys (tried to be inserted) are``    ``// distinct.``    ``if` `(x < root->data) {``        ``root->left = insert(root->left, x);``        ``root->lCount++;``    ``}` `    ``else` `if` `(x > root->data)``        ``root->right = insert(root->right, x);``    ``return` `root;``}` `// Function to find k'th smallest element in BST``// Here count denotes the number of nodes processed so far``Node* kthSmallest(Node* root, ``int` `k)``{``    ``// base case``    ``if` `(root == NULL)``        ``return` `NULL;``    ``int` `count = root->lCount + 1;``    ``if` `(count == k)``        ``return` `root;``    ``if` `(count > k)``        ``return` `kthSmallest(root->left, k);``    ``// else search in right subtree``    ``return` `kthSmallest(root->right, k - count);``}` `// main function``int` `main()``{``    ``Node* root = NULL;``    ``int` `keys[] = { 20, 8, 22, 4, 12, 10, 14 };``    ``int` `keys_size = ``sizeof``(keys) / ``sizeof``(keys);` `    ``for` `(``int` `i = 0; i < keys_size; i++)``        ``root = insert(root, keys[i]);``    ``int` `k = 4;``    ``Node* res = kthSmallest(root, k);``    ``if` `(res == NULL)``        ``printf``(``"There are less than k nodes in the BST"``);``    ``else``        ``printf``(``"K-th Smallest Element is %d"``, res->data);``    ``return` `0;``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Java

 `// A simple inorder traversal based Java program``// to find k-th smallest element in a BST.``import` `java.io.*;``import` `java.util.*;` `// A BST node``class` `Node {``    ``int` `data;``    ``Node left, right;``    ``int` `lCount;``    ``Node(``int` `x)``    ``{``        ``data = x;``        ``left = right = ``null``;``        ``lCount = ``0``;``    ``}``}` `class` `Gfg {``    ``// Recursive function to insert an key into BST``    ``public` `static` `Node insert(Node root, ``int` `x)``    ``{``        ``if` `(root == ``null``)``            ``return` `new` `Node(x);` `        ``// If a node is inserted in left subtree, then``        ``// lCount of this node is increased. For simplicity,``        ``// we are assuming that all keys (tried to be``        ``// inserted) are distinct.``        ``if` `(x < root.data) {``            ``root.left = insert(root.left, x);``            ``root.lCount++;``        ``}` `        ``else` `if` `(x > root.data)``            ``root.right = insert(root.right, x);``        ``return` `root;``    ``}` `    ``// Function to find k'th largest element in BST``    ``// Here count denotes the number of nodes processed so far``    ``public` `static` `Node kthSmallest(Node root, ``int` `k)``    ``{``        ``// base case``        ``if` `(root == ``null``)``            ``return` `null``;` `        ``int` `count = root.lCount + ``1``;``        ``if` `(count == k)``            ``return` `root;` `        ``if` `(count > k)``            ``return` `kthSmallest(root.left, k);` `        ``// else search in right subtree``        ``return` `kthSmallest(root.right, k - count);``    ``}` `    ``// main function``    ``public` `static` `void` `main(String args[])``    ``{``        ``Node root = ``null``;``        ``int` `keys[] = { ``20``, ``8``, ``22``, ``4``, ``12``, ``10``, ``14` `};` `        ``for` `(``int` `x : keys)``            ``root = insert(root, x);` `        ``int` `k = ``4``;``        ``Node res = kthSmallest(root, k);``        ``if` `(res == ``null``)``            ``System.out.println(``"There are less than k nodes in the BST"``);``        ``else``            ``System.out.println(``"K-th Smallest Element is "` `+ res.data);``    ``}``}` `// This code is contributed by Aditya Kumar (adityakumar129)`

## Python3

 `# A simple inorder traversal based Python3``# program to find k-th smallest element in a BST.` `# A BST node``class` `newNode:``    ` `    ``def` `__init__(``self``, x):``        ` `        ``self``.data ``=` `x``        ``self``.left ``=` `None``        ``self``.right ``=` `None``        ``self``.lCount ``=` `0` `# Recursive function to insert``# an key into BST``def` `insert(root, x):``    ` `    ``if` `(root ``=``=` `None``):``        ``return` `newNode(x)` `    ``# If a node is inserted in left subtree,``    ``# then lCount of this node is increased.``    ``# For simplicity, we are assuming that``    ``# all keys (tried to be inserted) are``    ``# distinct.``    ``if` `(x < root.data):``        ``root.left ``=` `insert(root.left, x)``        ``root.lCount ``+``=` `1` `    ``elif` `(x > root.data):``        ``root.right ``=` `insert(root.right, x);``        ` `    ``return` `root` `# Function to find k'th largest element``# in BST. Here count denotes the number``# of nodes processed so far``def` `kthSmallest(root, k):``    ` `    ``# Base case``    ``if` `(root ``=``=` `None``):``        ``return` `None``        ` `    ``count ``=` `root.lCount ``+` `1``    ` `    ``if` `(count ``=``=` `k):``        ``return` `root` `    ``if` `(count > k):``        ``return` `kthSmallest(root.left, k)` `    ``# Else search in right subtree``    ``return` `kthSmallest(root.right, k ``-` `count)` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``root ``=` `None``    ``keys ``=` `[ ``20``, ``8``, ``22``, ``4``, ``12``, ``10``, ``14` `]` `    ``for` `x ``in` `keys:``        ``root ``=` `insert(root, x)` `    ``k ``=` `4``    ``res ``=` `kthSmallest(root, k)``    ` `    ``if` `(res ``=``=` `None``):``        ``print``(``"There are less than k nodes in the BST"``)``    ``else``:``        ``print``(``"K-th Smallest Element is"``, res.data)``        ` `# This code is contributed by bgangwar59`

## C#

 `// A simple inorder traversal based C# program``// to find k-th smallest element in a BST.``using` `System;` `// A BST node``public` `class` `Node``{``    ``public` `int` `data;``    ``public` `Node left, right;``    ``public` `int` `lCount;``    ` `    ``public` `Node(``int` `x)``    ``{``        ``data = x;``        ``left = right = ``null``;``        ``lCount = 0;``    ``}``}` `class` `GFG{``    ` `// Recursive function to insert an key into BST``public` `static` `Node insert(Node root, ``int` `x)``{``    ``if` `(root == ``null``)``        ``return` `new` `Node(x);``        ` `    ``// If a node is inserted in left subtree,``    ``// then lCount of this node is increased.``    ``// For simplicity, we are assuming that``    ``// all keys (tried to be inserted) are``    ``// distinct.``    ``if` `(x < root.data)``    ``{``        ``root.left = insert(root.left, x);``        ``root.lCount++;``    ``}` `    ``else` `if` `(x > root.data)``        ``root.right = insert(root.right, x);``        ` `    ``return` `root;``}` `// Function to find k'th largest element``// in BST. Here count denotes the number``// of nodes processed so far``public` `static` `Node kthSmallest(Node root, ``int` `k)``{``    ` `    ``// Base case``    ``if` `(root == ``null``)``        ``return` `null``;` `    ``int` `count = root.lCount + 1;``    ``if` `(count == k)``        ``return` `root;` `    ``if` `(count > k)``        ``return` `kthSmallest(root.left, k);` `    ``// Else search in right subtree``    ``return` `kthSmallest(root.right, k - count);``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``Node root = ``null``;``    ``int``[] keys = { 20, 8, 22, 4, 12, 10, 14 };` `    ``foreach``(``int` `x ``in` `keys)``        ``root = insert(root, x);` `    ``int` `k = 4;``    ``Node res = kthSmallest(root, k);``    ` `    ``if` `(res == ``null``)``        ``Console.WriteLine(``"There are less "` `+``                          ``"than k nodes in the BST"``);``    ``else``        ``Console.WriteLine(``"K-th Smallest"` `+``                          ``" Element is "` `+ res.data);``}``}` `// This code is contributed by aashish1995`

## Javascript

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Output:

`K-th Smallest Element is 12`

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

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