Height of Binary Tree in DSA Java - Time & Space Complexity

Time Complexity: Height of Binary Tree

O(n)

Understanding Time Complexity

We want to understand how the time to find the height of a binary tree grows as the tree gets bigger.

How does the number of steps change when the tree has more nodes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public int height(TreeNode root) {
    if (root == null) return 0;
    int leftHeight = height(root.left);
    int rightHeight = height(root.right);
    return 1 + Math.max(leftHeight, rightHeight);
}
    

This code finds the height of a binary tree by checking the height of left and right subtrees recursively.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls visiting each node once.
• How many times: Once per node in the tree.

How Execution Grows With Input

As the number of nodes grows, the function visits each node once to calculate height.

Input Size (n)	Approx. Operations
10	About 10 calls
100	About 100 calls
1000	About 1000 calls

Pattern observation: The number of steps grows roughly in direct proportion to the number of nodes.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the height grows linearly with the number of nodes in the tree.

Common Mistake

[X] Wrong: "The height calculation only depends on the longest path, so it takes O(log n) time."

[OK] Correct: Even though height is about the longest path, the function must visit every node to be sure, so it takes time proportional to all nodes, not just the longest path.

Interview Connect

Understanding this helps you explain how recursive tree algorithms work and how their time depends on tree size, a common topic in interviews.

Self-Check

"What if the tree was a balanced binary search tree? How would the time complexity change?"

Practice

1. What is the height of a binary tree with only one node (the root)?

easy

A. Undefined

B. 1

C. 2

D. 0

Solution

1. Step 1: Understand height definition

   The height of a binary tree is the number of edges on the longest path from the root to a leaf.

2. Step 2: Apply to single node tree

   With only one node (root), the longest path has length 0 edges.

3. Final Answer:

   0 -> Option D

4. Quick Check:

   Single node tree height = 0 [OK]

Hint: Height counts edges on longest root-to-leaf path [OK]

Common Mistakes:

• Thinking height is 1 for single node
• Confusing height with number of nodes
• Assuming height undefined for single node

2. Which of the following Java method signatures correctly defines a recursive method to find the height of a binary tree node?

easy

A. public int height(Node root)

B. int height(Node root) void

C. public void height(Node root) returns int

D. height(Node root) int public

Solution

1. Step 1: Check Java method syntax

   Java methods must specify access modifier, return type, method name, and parameters correctly.

2. Step 2: Validate each option

   public int height(Node root) follows correct syntax: public int height(Node root). Others have syntax errors.

3. Final Answer:

   public int height(Node root) -> Option A

4. Quick Check:

   Correct Java method signature = public int height(Node root) [OK]

Hint: Java method: access + return type + name + parameters [OK]

Common Mistakes:

• Wrong order of return type and name
• Missing return type
• Using void with return value

3. What will be the output of the following Java code snippet?

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

public class Test {
  public static int height(Node root) {
    if (root == null) return 0;
    int leftHeight = height(root.left);
    int rightHeight = height(root.right);
    return Math.max(leftHeight, rightHeight) + 1;
  }

  public static void main(String[] args) {
    Node root = new Node(1);
    root.left = new Node(2);
    root.right = new Node(3);
    root.left.left = new Node(4);
    System.out.println(height(root));
  }
}

medium

A. 1

B. 3

C. 2

D. 4

Solution

1. Step 1: Visualize the tree structure

   The tree has root 1, left child 2, right child 3, and left child of 2 is 4. The longest path is 1->2->4.

2. Step 2: Calculate height recursively

   Height is max depth: left subtree height = 2 (nodes 2->4), right subtree height = 1 (node 3), so total height = max(2,1)+1 = 3.

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Longest path length = 3 [OK]

Hint: Height = longest root-to-leaf path node count [OK]

Common Mistakes:

• Counting edges instead of nodes
• Ignoring null child base case
• Misreading tree structure

4. Identify the error in the following Java method to find the height of a binary tree:

public int height(Node root) {
  if (root == null) {
    return -1;
  }
  int leftHeight = height(root.left);
  int rightHeight = height(root.right);
  return Math.max(leftHeight, rightHeight);
}

medium

A. The method should not use recursion.

B. The method should return the sum of left and right heights.

C. The base case should return 0, not -1.

D. The method should return the minimum of left and right heights.

Solution

1. Step 1: Check base case return value

   Height of empty tree (null) should be 0 or consistent with counting edges. Returning -1 causes incorrect height calculation.

2. Step 2: Check return statement

   Method returns max of left and right heights but misses adding 1 for current node, so height is underestimated.

3. Final Answer:

   The base case should return 0, not -1. -> Option C

4. Quick Check:

   Base case must return 0 for correct height [OK]

Hint: Base case return 0, add 1 after max height [OK]

Common Mistakes:

• Returning -1 in base case
• Not adding 1 to max height
• Returning sum instead of max

5. Given a binary tree where each node stores an integer, write a method to find the height of the tree and then print all nodes at the maximum height level. Which approach correctly combines these tasks?

hard

A. First find height recursively, then use a level-order traversal to print nodes at that height.

B. Print nodes during height calculation without storing height separately.

C. Use in-order traversal to print nodes at maximum height level directly.

D. Calculate height by counting nodes and print nodes at root only.

Solution

1. Step 1: Find height recursively

   Use a recursive method to find the height of the tree, which gives the maximum depth level.

2. Step 2: Use level-order traversal

   Perform a breadth-first traversal (level-order) and print nodes only at the level equal to the height found.

3. Final Answer:

   First find height recursively, then use a level-order traversal to print nodes at that height. -> Option A

4. Quick Check:

   Separate height calculation + level-order print = First find height recursively, then use a level-order traversal to print nodes at that height. [OK]

Hint: Find height first, then print nodes at that level [OK]

Common Mistakes:

• Trying to print during height recursion
• Using in-order traversal for level nodes
• Printing root only ignoring height