Height of Binary Tree in DSA Typescript - 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.

The question is: 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 heightOfBinaryTree(root: TreeNode | null): number {
  if (root === null) return 0;
  const leftHeight = heightOfBinaryTree(root.left);
  const rightHeight = heightOfBinaryTree(root.right);
  return Math.max(leftHeight, rightHeight) + 1;
}
    

This code finds the height by checking the height of left and right subtrees recursively and returning the larger one plus one.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls visiting each node once.
• How many times: Each node in the tree is visited exactly one time.

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 visits
100	About 100 visits
1000	About 1000 visits

Pattern observation: The time grows linearly as the tree size increases.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the height grows in direct proportion to the number of nodes in the tree.

Common Mistake

[X] Wrong: "The height calculation only depends on the longest path, so it should be faster than visiting all nodes."

[OK] Correct: Even though height depends on the longest path, the function must check every node to be sure no taller path exists, so it visits all nodes.

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 balanced versus very skewed? How would the time complexity change?"

Practice

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

easy

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

B. The total number of nodes in the tree

C. The number of leaf nodes in the tree

D. The number of nodes at the root level

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 A

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 TypeScript function signature correctly defines a recursive function to find the height of a binary tree node?

easy

A. function height(node: TreeNode | null): number

B. function height(node: number): number

C. function height(): void

D. function height(node: TreeNode): string

Solution

1. Step 1: Identify the correct parameter type for a tree node

   The function must accept a node or null (for empty subtree), so type is TreeNode | null.

2. Step 2: Confirm the return type is number

   The height is a numeric value, so the return type must be number.

3. Final Answer:

   function height(node: TreeNode | null): number -> Option A

4. Quick Check:

   Parameter: TreeNode|null, Return: number [OK]

Hint: Height function needs node or null input and returns number [OK]

Common Mistakes:

• Using wrong parameter type like number instead of node
• Returning void or string instead of number
• Missing null to handle empty subtree

3. What is the height of the following binary tree?

    1
   / \
  2   3
 / 
4  

Consider height as number of edges on longest path from root to leaf.

medium

A. 3

B. 4

C. 1

D. 2

Solution

1. Step 1: Identify all root-to-leaf paths and count edges

   Paths: 1->2->4 has 2 edges, 1->3 has 1 edge.

2. Step 2: Find the longest path edge count

   Longest path is 1->2->4 with 2 edges.

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Longest path edges = 2 [OK]

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

Common Mistakes:

• Counting nodes instead of edges
• Choosing shorter path length
• Confusing height with tree depth

4. Identify the error in this TypeScript function to find binary tree height:

function height(node: TreeNode | null): number {
  if (!node) return 0;
  return Math.max(height(node.left), height(node.right));
}

medium

A. Returns 0 for null node instead of -1

B. Uses Math.min instead of Math.max

C. Missing +1 to count current node edge

D. Function should return void

Solution

1. Step 1: Analyze base case return value

   Returning 0 for null means empty subtree height is 0, which is acceptable if counting edges from nodes.

2. Step 2: Check recursive return statement

   Function returns max height of children but misses adding 1 for current edge from node to child.

3. Final Answer:

   Missing +1 to count current node edge -> Option C

4. Quick Check:

   Height = max child height + 1 edge [OK]

Hint: Add +1 after max to include current node edge [OK]

Common Mistakes:

• Forgetting to add 1 for current node edge
• Returning wrong base case value
• Using min instead of max for height

5. Given a binary tree where each node has a value, write a function to find the height of the tree but only counting nodes with even values. What is the height of this tree?

    2
   / \
  3   4
 /   / \
6   5   8

hard

A. 0

B. 2

C. 1

D. 3

Solution

1. Step 1: Identify nodes with even values and their connections

   Even nodes: 2 (root), 4 (right child), 6 (left-left child), 8 (right-right child). Paths counting only even nodes:
• 2 -> 4 -> 8 (edges = 2)
• 2 -> 3 (3 is odd, so path ends at 2)
• 2 -> 3 -> 6 (6 is even but 3 is odd, so path breaks at 3)

2. Step 2: Calculate height counting only edges between even nodes

   Longest path with even nodes is 2 -> 4 -> 8 with 2 edges.

3. Final Answer:

   2 -> Option B

4. Quick Check:

   Longest even-valued node path edges = 2 [OK]

Hint: Count edges only between even-valued nodes [OK]

Common Mistakes:

• Counting odd nodes in path
• Counting nodes instead of edges
• Ignoring breaks caused by odd nodes