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

Time Complexity: Height of Binary Tree

O(n)

Understanding Time Complexity

We want to know how long it takes to find the height of a binary tree as the tree grows bigger.

How does the time needed change when the tree has more nodes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function height(node) {
  if (node === null) return 0;
  const leftHeight = height(node.left);
  const rightHeight = height(node.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, so the work grows directly with the number of nodes.

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

Pattern observation: The time grows linearly with the number of nodes.

Final Time Complexity

Time Complexity: O(n)

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

Common Mistake

[X] Wrong: "The height function only checks one path, so it runs in constant time."

[OK] Correct: The function must visit every node to find the height, not just one path, so it takes time proportional to all nodes.

Interview Connect

Understanding how recursive tree functions scale helps you explain your solutions clearly and shows you know how to handle common data structures.

Self-Check

"What if the tree was balanced vs very unbalanced? How would that affect the time complexity?"

Practice

1. What does the height of a binary tree represent?

easy

A. The number of children of the root node

B. The total number of nodes in the tree

C. The number of leaf nodes in the tree

D. The number of edges on the longest path from root to a leaf

Solution

1. Step 1: Understand the definition of height in a binary tree

   The height is the longest path from the root node down to the farthest leaf node, counted by edges.

2. Step 2: Differentiate height from other tree properties

   Number of nodes or leaf nodes are different concepts; height specifically counts edges on the longest path.

3. Final Answer:

   The number of edges on the longest path from root to a leaf -> Option D

4. Quick Check:

   Height = longest root-to-leaf edge count [OK]

Hint: Height counts edges, not nodes, on longest root-to-leaf path [OK]

Common Mistakes:

• Confusing height with total nodes count
• Counting nodes instead of edges for height
• Thinking height equals number of leaf nodes

2. Which of the following is the correct base case to find the height of a binary tree recursively in JavaScript?

easy

A. if (node === null) return -1;

B. if (node === null) return 0;

C. if (node === undefined) return 1;

D. if (node === undefined) return 0;

Solution

1. Step 1: Understand the base case for empty subtree

   When the node is null (no node), height is defined as -1 to count edges properly.

2. Step 2: Why -1 instead of 0?

   Returning -1 for null ensures leaf nodes have height 0 (0 edges below them), matching edge count definition.

3. Final Answer:

   if (node === null) return -1; -> Option A

4. Quick Check:

   Empty node height = -1 for correct edge count [OK]

Hint: Use -1 for null node to count edges correctly [OK]

Common Mistakes:

• Returning 0 for null node causes off-by-one height
• Checking undefined instead of null for nodes
• Confusing node absence with leaf node

3. What is the output of the following code snippet?

class Node {
  constructor(val) {
    this.val = val;
    this.left = null;
    this.right = null;
  }
}

function height(node) {
  if (node === null) return -1;
  return 1 + Math.max(height(node.left), height(node.right));
}

const root = new Node(10);
root.left = new Node(5);
root.right = new Node(15);
root.left.left = new Node(3);
root.left.left.left = new Node(1);

console.log(height(root));

medium

A. 5

B. 3

C. 4

D. 2

Solution

1. Step 1: Trace the tree structure and count edges

   The longest path is from root (10) to left-left-left (1): edges are 10->5, 5->3, 3->1, total 3 edges.

2. Step 2: Calculate height using the function logic

   Function returns 1 + max height of children. For leaf node (1), height = 0. So total height = 3 edges means function returns 3.

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Longest path edges = 3 [OK]

Hint: Count edges on longest path, not nodes [OK]

Common Mistakes:

• Counting nodes instead of edges for height
• Misreading tree structure and missing deepest node
• Confusing return value for leaf node height

4. Find the error in this height function and select the correct fix:

function height(node) {
  if (node === null) return 0;
  return Math.max(height(node.left), height(node.right));
}

medium

A. Change base case to return -1 instead of 0

B. Return sum instead of max of left and right heights

C. Add +1 before Math.max to count current node

D. No error, function is correct

Solution

1. Step 1: Analyze the base case and recursive return

   Base case returns 0 for null, which is acceptable if counting nodes, but function misses adding 1 for current node.

2. Step 2: Identify missing increment for current node

   Function returns max height of children but does not add 1 for current node, so height is underestimated by 1.

3. Final Answer:

   Add +1 before Math.max to count current node -> Option C

4. Quick Check:

   Height = 1 + max(child heights) [OK]

Hint: Add +1 to include current node in height [OK]

Common Mistakes:

• Forgetting to add 1 for current node
• Changing base case unnecessarily
• Returning sum instead of max height

5. Given a binary tree where some nodes have only one child, how does the height calculation change if you want to count the number of nodes on the longest path instead of edges? Choose the correct modification to the recursive height function.

function height(node) {
  if (node === null) return ?;
  return ? + Math.max(height(node.left), height(node.right));
}

hard

A. Return 0 for null; add 1 before Math.max to count nodes

B. Return -1 for null; add 1 before Math.max to count nodes

C. Return 1 for null; add 0 before Math.max to count nodes

D. Return 0 for null; add 0 before Math.max to count nodes

Solution

1. Step 1: Understand difference between counting edges and nodes

   Counting nodes means height of empty tree is 0, and each node adds 1 to path length.

2. Step 2: Adjust base case and recursive addition accordingly

   Return 0 for null node (no nodes), and add 1 for current node to count nodes on path.

3. Final Answer:

   Return 0 for null; add 1 before Math.max to count nodes -> Option A

4. Quick Check:

   Height counts nodes: base=0, add 1 per node [OK]

Hint: Use 0 base and add 1 to count nodes, not edges [OK]

Common Mistakes:

• Using -1 base for node count causes off-by-one
• Not adding 1 for current node
• Confusing edges count with nodes count