Maximum Depth of Binary Tree

Last Updated : 03 Apr, 2025

Given a binary tree, the task is to find the maximum depth of the tree. The maximum depth or height of the tree is the number of edges in the tree from the root to the deepest node.

Examples:

Input:

Output: 2
Explanation: The longest path from the root (node 12) goes through node 8 to node 5, which has 2 edges.

Input:

Output: 3
Explanation: The longest path from the root (node 1) to a leaf node 6 with 3 edge.

Table of Content

[Approach 1] Using Recursion – O(n) Time and O(h) Space

The idea is to recursively calculate the height of the left and the right subtrees of a node and then find height to the node as max of the heights of two children plus 1.

Algorithm:

If the tree is empty, return -1.
Otherwise, do the following:
- Get the height of the left subtree recursively, i.e., call height(node->left).
- Get the height of the right subtree recursively, i.e., call height(node->right).
- Compute the maximum of the heights of the left and right subtrees and add 1 to it for the current node.
- height = max(height of left subtree, height of right subtree) + 1.
Return the height.

C++

#include <iostream> using namespace std;  class Node { public:     int data;     Node *left;     Node *right;      Node(int val) {         data = val;         left = nullptr;         right = nullptr;     } };  // Returns height which is the number of edges // along the longest path from the root node down  // to the farthest leaf node. int height(Node *root) {     if (root == nullptr)         return -1;      // compute the height of left and right subtrees     int lHeight = height(root->left);     int rHeight = height(root->right);      return max(lHeight, rHeight) + 1; }  int main() {        // Representation of the input tree:     //     12     //    /  \     //   8   18     //  / \     // 5   11     Node *root = new Node(12);     root->left = new Node(8);     root->right = new Node(18);     root->left->left = new Node(5);     root->left->right = new Node(11);          cout << height(root);          return 0; }

#include <stdio.h> #include <stdlib.h>  struct Node {     int data;     struct Node* left;     struct Node* right; };   // Returns "height" which is the number of edges  // along the longest path from the root node down  // to the farthest leaf node. int height(struct Node* root) {     if (root == NULL)         return -1;      // compute the height of left and right subtrees     int lHeight = height(root->left);     int rHeight = height(root->right);      return (lHeight > rHeight ? lHeight : rHeight) + 1; }  struct Node* createNode(int val) {     struct Node* node        		= (struct Node*)malloc(sizeof(struct Node));     node->data = val;     node->left = NULL;     node->right = NULL;     return node; }  int main() {        // Representation of the input tree:     //     12     //    /  \     //   8   18     //  / \     // 5   11     struct Node* root = createNode(12);     root->left = createNode(8);     root->right = createNode(18);     root->left->left = createNode(5);     root->left->right = createNode(11);          printf("%d\n", height(root));          return 0; }

Java

class Node {     int data;     Node left, right;      Node(int val) {         data = val;         left = null;         right = null;     } }  class GfG {      // Returns height which is the number of edges     // along the longest path from the root node down      // to the farthest leaf node.     static int height(Node root) {         if (root == null)             return -1;          // compute the height of left and right subtrees         int lHeight = height(root.left);         int rHeight = height(root.right);          return Math.max(lHeight, rHeight) + 1;     }      public static void main(String[] args) {          // Representation of the input tree:         //     12         //    /  \         //   8   18         //  / \         // 5   11         Node root = new Node(12);         root.left = new Node(8);         root.right = new Node(18);         root.left.left = new Node(5);         root.left.right = new Node(11);          System.out.println(height(root));     } }

Python

class Node:     def __init__(self, val):         self.data = val         self.left = None         self.right = None  # Returns height which is the number of edges # along the longest path from the root node down  # to the farthest leaf node. def height(root):     if root is None:         return -1      # compute the height of left and right subtrees     lHeight = height(root.left)     rHeight = height(root.right)      return max(lHeight, rHeight) + 1  if __name__ == "__main__":        # Representation of the input tree:     #     12     #    /  \     #   8   18     #  / \     # 5   11     root = Node(12)     root.left = Node(8)     root.right = Node(18)     root.left.left = Node(5)     root.left.right = Node(11)      print(height(root))

using System;  class Node {     public int data;     public Node left, right;      public Node(int val) {         data = val;         left = null;         right = null;     } }  class GfG {      // Returns height which is the number of edges     // along the longest path from the root node down      // to the farthest leaf node.     static int height(Node root) {         if (root == null)             return -1;          // compute the height of left and right subtrees         int lHeight = height(root.left);         int rHeight = height(root.right);          return Math.Max(lHeight, rHeight) + 1;     }      static void Main(string[] args) {          // Representation of the input tree:         //     12         //    /  \         //   8   18         //  / \         // 5   11         Node root = new Node(12);         root.left = new Node(8);         root.right = new Node(18);         root.left.left = new Node(5);         root.left.right = new Node(11);          Console.WriteLine(height(root));     } }

JavaScript

class Node {     constructor(val) {         this.data = val;         this.left = null;         this.right = null;     } }  // Returns height which is the number of edges // along the longest path from the root node down  // to the farthest leaf node. function height(root) {     if (root === null) {         return -1;     }      // compute the height of left and right subtrees     let lHeight = height(root.left);     let rHeight = height(root.right);      return Math.max(lHeight, rHeight) + 1; }  // Driver Code  // Representation of the input tree: //     12 //    /  \ //   8   18 //  / \ // 5   11 let root = new Node(12); root.left = new Node(8); root.right = new Node(18); root.left.left = new Node(5); root.left.right = new Node(11);  console.log(height(root));

Output

Below is the illustration of above approach:

Example: Consider the following tree –

height(’12’) = max(height(‘8′), height(’18’)) + 1 = 1 + 1 = 2
because recursively:
height(‘8’) = max(height(‘5′), height(’11’)) + 1 = 0 + 1 = 1
height(’18’) = max(height(NULL), height(‘NULL’)) + 1 = (-1) + 1 = 0
height(“5”) = max(height(NULL), height(‘NULL’)) + 1 = (-1) + 1 = 0
height(“11”) = max(height(NULL), height(‘NULL’)) + 1 = (-1) + 1 = 0

[Approach 2] Level Order Traversal – O(n) Time and O(n) Space

If we take a closer look at the breadth first traversal, we can notice that after we process the last node of the current level, the queue contains all the nodes of next level. This is true from the root (or first level) So we get the size of the queue in a loop to count the nodes of every level.

Algorithm:

Initialize an empty queue q to store nodes of the tree and a variable depth to keep track of the number of levels.
Push the root node into the queue q.
While the queue is not empty:
- Store the number of nodes at the current level as levelSize (the size of the queue).
- For each node in the current level (loop from 0 to levelSize):
  - Dequeue the front element from the queue (i.e., process the node).
  - If the node has a left child, enqueue it.
  - If the node has a right child, enqueue it.
- After processing all nodes at the current level, increment the depth.
Return depth - 1 as the number of edges encountered will be one less than number of nodes.

C++

#include <iostream> #include <queue> using namespace std;  class Node { public:     int data;     Node *left, *right;     Node(int val) {         data = val;         left = nullptr;         right = nullptr;     } };  int height(Node* root) {     if (!root) return 0;      // Initializing a queue to traverse     // the tree level by level     queue<Node*> q;     q.push(root);     int depth = 0;      // Loop until the queue is empty     while (!q.empty()) {         int levelSize = q.size();          // Traverse all nodes at the current level         for (int i = 0; i < levelSize; i++) {             Node* curr = q.front();             q.pop();             if (curr->left) q.push(curr->left);             if (curr->right) q.push(curr->right);         }          // Increment depth after traversing each level         depth++;     }     return depth - 1; }  int main() {        // Representation of the input tree:     //     12     //    /  \     //   8   18     //  / \     // 5   11     Node *root = new Node(12);     root->left = new Node(8);     root->right = new Node(18);     root->left->left = new Node(5);     root->left->right = new Node(11);          cout << height(root);          return 0; }

Java

import java.util.LinkedList; import java.util.Queue;  class Node {     int data;     Node left, right;      Node(int val) {         data = val;         left = null;         right = null;     } }  class GfG {     static int height(Node root) {         if (root == null) return 0;          // Initializing a queue to traverse         // the tree level by level         Queue<Node> q = new LinkedList<>();         q.add(root);         int depth = 0;          // Loop until the queue is empty         while (!q.isEmpty()) {             int levelSize = q.size();              // Traverse all nodes at the current level             for (int i = 0; i < levelSize; i++) {                 Node curr = q.poll();                  if (curr.left != null) q.add(curr.left);                 if (curr.right != null) q.add(curr.right);             }              // Increment height after traversing each level             depth++;         }         return depth - 1;     }      public static void main(String[] args) {          // Representation of the input tree:         //     12         //    /  \         //   8   18         //  / \         // 5   11         Node root = new Node(12);         root.left = new Node(8);         root.right = new Node(18);         root.left.left = new Node(5);         root.left.right = new Node(11);          System.out.println(height(root));     } }

Python

from collections import deque  class Node:     def __init__(self, data):         self.data = data         self.left = None         self.right = None  # Function to find the height of the tree def height(root):     if root is None:         return 0      # Initializing a queue to traverse     # the tree level by level     q = deque([root])     depth = 0      # Loop until the queue is empty     while q:         levelSize = len(q)          # Traverse all nodes at the current level         for _ in range(levelSize):             curr = q.popleft()              if curr.left:                 q.append(curr.left)             if curr.right:                 q.append(curr.right)          # Increment depth after traversing each level         depth += 1      return depth - 1  if __name__ == "__main__":        # Representation of the input tree:     #     12     #    /  \     #   8   18     #  / \     # 5   11     root = Node(12)     root.left = Node(8)     root.right = Node(18)     root.left.left = Node(5)     root.left.right = Node(11)      print(height(root))

using System; using System.Collections.Generic;  class Node {     public int data;     public Node left, right;      public Node(int x) {         data = x;         left = null;         right = null;     } }  class GfG {        static int height(Node root) {         if (root == null) {             return 0;         }          // Initializing a queue to traverse         // the tree level by level         Queue<Node> q = new Queue<Node>();         q.Enqueue(root);         int depth = 0;          // Loop until the queue is empty         while (q.Count > 0) {             int levelSize = q.Count;              // Traverse all nodes at the current level             for (int i = 0; i < levelSize; i++) {                 Node curr = q.Dequeue();                  if (curr.left != null) {                     q.Enqueue(curr.left);                 }                 if (curr.right != null) {                     q.Enqueue(curr.right);                 }             }              // Increment depth after traversing            	// a level             depth++;         }         return depth - 1;     }      static void Main(string[] args) {          // Representation of the input tree:         //     12         //    /  \         //   8   18         //  / \         // 5   11         Node root = new Node(12);         root.left = new Node(8);         root.right = new Node(18);         root.left.left = new Node(5);         root.left.right = new Node(11);          Console.WriteLine(height(root));     } }

JavaScript

class Node {     constructor(data) {         this.data = data;         this.left = null;         this.right = null;     } }  // Function to find the height of the tree function height(root) {     if (root === null) {         return 0;     }      // Initializing a queue to traverse     // the tree level by level     let queue = [root];     let depth = 0;      // Loop until the queue is empty     while (queue.length > 0) {         let levelSize = queue.length;          // Traverse all nodes at the current level         for (let i = 0; i < levelSize; i++) {             let curr = queue.shift();              if (curr.left) {                 queue.push(curr.left);             }             if (curr.right) {                 queue.push(curr.right);             }         }          // Increment depth after traversing a level         depth++;	     }     return depth - 1; }   // Representation of the input tree: //     12 //    /  \ //   8   18 //  / \ // 5   11 let root = new Node(12); root.left = new Node(8); root.right = new Node(18); root.left.left = new Node(5); root.left.right = new Node(11);  console.log(height(root));

Output

Insertion in a Binary Tree in level order

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Maximum Depth of Binary Tree

[Approach 1] Using Recursion – O(n) Time and O(h) Space

[Approach 2] Level Order Traversal – O(n) Time and O(n) Space

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