Height of Binary Tree in DSA C# - Time & Space Complexity

Time Complexity: Height of Binary Tree

O(n)

Understanding Time Complexity

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

How does the work increase when the tree has more nodes?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public class Node {
    public int Value;
    public Node Left, Right;
}

int Height(Node 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

Each node is visited 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 work grows linearly as the tree size increases.

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 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 why visiting all nodes is often necessary.

Self-Check

"What if the tree was a linked list (all nodes have only one child)? 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 children of the root node

Solution

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

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

2. Step 2: Compare with other options

   Other options describe different properties: total nodes, leaf count, or root children, which are not height.

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 or leaves [OK]

Common Mistakes:

• Confusing height with total nodes
• Counting nodes instead of edges
• Thinking height is number of leaves

2. Which of the following C# method signatures correctly defines a recursive function to find the height of a binary tree node named root?

easy

A. public int Height(Node root, int height)

B. public void Height(Node root)

C. public int Height()

D. public int Height(Node root)

Solution

1. Step 1: Identify correct return type and parameters

   The height function must return an integer representing height, so return type is int. It needs the root node as input, so parameter is Node root.

2. Step 2: Check options for matching signature

   public int Height(Node root) matches: returns int and takes Node root. public void Height(Node root) returns void (wrong), C has no parameters (wrong), D has extra parameter not needed for basic height.

3. Final Answer:

   public int Height(Node root) -> Option D

4. Quick Check:

   Return int and take root node parameter [OK]

Hint: Height returns int and takes root node as parameter [OK]

Common Mistakes:

• Using void return type
• Missing root parameter
• Adding unnecessary parameters

3. What is the output of the following C# code snippet that calculates the height of a binary tree?

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

int Height(Node root) {
  if (root == null) return -1;
  int leftHeight = Height(root.left);
  int rightHeight = Height(root.right);
  return Math.Max(leftHeight, rightHeight) + 1;
}

// Tree:
//     1
//    / \
//   2   3
//  /
// 4

Node root = new Node(1);
root.left = new Node(2);
root.right = new Node(3);
root.left.left = new Node(4);

Console.WriteLine(Height(root));

medium

A. 1

B. 3

C. 2

D. 4

Solution

1. Step 1: Understand the height calculation logic

   Height returns -1 for null nodes. For leaf nodes, height = 0. Height is max of left and right subtree heights plus 1.

2. Step 2: Calculate height for given tree

   Node 4: left and right null, height = 0.
   Node 2: left height 0 (node 4), right null (-1), max=0 +1=1.
   Node 3: leaf, height=0.
   Root 1: left height 1, right height 0, max=1 +1=2.

3. Final Answer:

   2 -> Option C

4. Quick Check:

   Height of root = 2 edges [OK]

Hint: Height counts edges; leaf node height is 0 [OK]

Common Mistakes:

• Counting nodes instead of edges
• Returning 0 for null instead of -1
• Miscomputing max height

4. Identify the error in this C# method to find the height of a binary tree and select the fix:

int Height(Node root) {
  if (root == null) return 0;
  int leftHeight = Height(root.left);
  int rightHeight = Height(root.right);
  return Math.Max(leftHeight, rightHeight);
}

medium

A. Change base case return from 0 to -1

B. Add +1 to the return statement after Math.Max

C. Remove recursive calls to left and right

D. Change Math.Max to Math.Min

Solution

1. Step 1: Analyze base case and return statement

   Base case returns 0 for null, which means height of empty tree is 0. The recursive calls get left and right heights correctly.

2. Step 2: Check final return calculation

   Height should be max of left and right heights plus 1 for current node. The code misses +1, so height is underestimated.

3. Final Answer:

   Add +1 to the return statement after Math.Max -> Option B

4. Quick Check:

   Height = max(left, right) + 1 [OK]

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

Common Mistakes:

• Forgetting +1 in return
• Changing base case incorrectly
• Using min 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 return the sum of all node values at the maximum height level. Which approach correctly solves this problem?

hard

A. First find height, then traverse tree again to sum nodes at that height

B. Sum all nodes during height calculation recursively

C. Return height multiplied by root node value

D. Sum all leaf nodes regardless of height

Solution

1. Step 1: Understand problem requirements

   We need the height of the tree and sum of nodes at that height level (deepest level).

2. Step 2: Choose correct approach

   First find height, then traverse tree again to sum nodes at that height: Find height first, then traverse again to sum nodes at that level. This separates concerns and is clear.
   Sum all nodes during height calculation recursively tries to sum during height calculation, but height is computed bottom-up and summing nodes at max height requires separate traversal.
   Return height multiplied by root node value and D do not meet problem requirements.

3. Final Answer:

   First find height, then traverse tree again to sum nodes at that height -> Option A

4. Quick Check:

   Two-pass approach: height then sum at height [OK]

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

Common Mistakes:

• Trying to sum nodes during height recursion
• Multiplying height by root value
• Summing all leaf nodes ignoring height