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

Time Complexity: Height of Binary Tree

O(n)

Understanding Time Complexity

We want to understand 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.

func height(root *Node) int {
  if root == nil {
    return 0
  }
  leftHeight := height(root.Left)
  rightHeight := height(root.Right)
  if leftHeight > rightHeight {
    return leftHeight + 1
  }
  return rightHeight + 1
}
    

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 operations grows directly with 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 should be O(log n)."

[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.

Interview Connect

Understanding this helps you explain how recursive tree algorithms work and why visiting all nodes is often necessary.

Self-Check

"What if we stored the height at each node during insertion? How would the time complexity of finding height change?"

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: Compare options with the definition

   Only The number of edges on the longest path from root to a leaf correctly describes the height as the number of edges on the longest root-to-leaf 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 edges [OK]

Hint: Height counts edges, not nodes, from root to deepest leaf [OK]

Common Mistakes:

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

2. Which Go function signature correctly defines a recursive function to find the height of a binary tree node?

easy

A. func height(root Node) string

B. func height(root *Node) int

C. func height(root *Node) bool

D. func height(root Node) float64

Solution

1. Step 1: Identify correct parameter and return types for height function

   The function should accept a pointer to a Node and return an integer representing height.

2. Step 2: Match options with correct signature

   func height(root *Node) int matches: func height(root *Node) int. Others have wrong parameter types or return types.

3. Final Answer:

   func height(root *Node) int -> Option B

4. Quick Check:

   Pointer input and int output for height [OK]

Hint: Height returns int; input is pointer to Node [OK]

Common Mistakes:

• Using value instead of pointer for root
• Returning wrong data type like bool or string
• Incorrect function parameter types

3. What is the output of this Go code snippet that calculates the height of a binary tree?

package main

import "fmt"

type Node struct {
    left, right *Node
}

func height(root *Node) int {
    if root == nil {
        return 0
    }
    leftHeight := height(root.left)
    rightHeight := height(root.right)
    if leftHeight > rightHeight {
        return leftHeight + 1
    }
    return rightHeight + 1
}

func main() {
    root := &Node{
        left: &Node{},
        right: &Node{
            right: &Node{},
        },
    }
    fmt.Println(height(root))
}

medium

A. 3

B. 2

C. 4

D. 1

Solution

1. Step 1: Analyze the tree structure

   The root has left child (leaf) and right child with its own right child (leaf). The longest path is root -> right -> right, which has 3 nodes and 2 edges.

2. Step 2: Calculate height as edges + 1

   The function returns 1 + max height of children. The longest path has 2 edges, so height = 3.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Height counts edges + 1 for nodes [OK]

Hint: Count longest path edges + 1 for height [OK]

Common Mistakes:

• Counting nodes instead of edges
• Ignoring nil child nodes
• Confusing left and right subtree heights

4. Identify the bug in this Go function that calculates the height of a binary tree:

func height(root *Node) int {
    if root == nil {
        return -1
    }
    leftHeight := height(root.left)
    rightHeight := height(root.right)
    if leftHeight > rightHeight {
        return leftHeight + 1
    }
    return rightHeight + 1
}

medium

A. Returning -1 for nil node causes incorrect height calculation

B. Missing base case for leaf nodes

C. Using pointers incorrectly for root

D. Not handling empty tree separately

Solution

1. Step 1: Check base case return value

   The function returns -1 for nil nodes, which causes height to be off by one compared to standard definition.

2. Step 2: Understand correct base case for height

   Height of empty tree should be 0 or 0 edges; returning -1 leads to negative heights and incorrect results.

3. Final Answer:

   Returning -1 for nil node causes incorrect height calculation -> Option A

4. Quick Check:

   Base case should return 0 for nil [OK]

Hint: Base case for nil must return 0, not -1 [OK]

Common Mistakes:

• Returning -1 instead of 0 for nil
• Confusing base case with leaf node
• Not adding 1 after recursive calls

5. Given a binary tree where each node stores an integer, write a recursive Go function to find the height of the tree and also return the sum of all node values at the deepest level. Which approach correctly combines these tasks?

hard

A. Use iterative BFS to find height and sum nodes at last level in one pass

B. Calculate height first, then traverse tree again to sum deepest level nodes separately

C. Use a recursive function that returns both height and sum at deepest level by comparing left and right subtree results

D. Use recursion to sum all nodes and ignore height calculation

Solution

1. Step 1: Understand combined task requirements

   We need height and sum of nodes at deepest level in one function efficiently.

2. Step 2: Evaluate approaches

   Use a recursive function that returns both height and sum at deepest level by comparing left and right subtree results uses recursion returning height and sum together, avoiding multiple traversals.

3. Step 3: Compare other options

   Calculate height first, then traverse tree again to sum deepest level nodes separately does two traversals, less efficient. Use iterative BFS to find height and sum nodes at last level in one pass is valid but iterative BFS is not recursive. Use recursion to sum all nodes and ignore height calculation ignores height, so incomplete.

4. Final Answer:

   Use a recursive function that returns both height and sum at deepest level by comparing left and right subtree results -> Option C

5. Quick Check:

   Combine height and sum in one recursive pass [OK]

Hint: Return height and sum together recursively to avoid extra traversal [OK]

Common Mistakes:

• Doing two separate traversals
• Ignoring sum at deepest level
• Using iterative approach when recursion asked