Tree Terminology Root Leaf Height Depth Level in DSA Go - Time & Space Complexity

Time Complexity: Tree Terminology Root Leaf Height Depth Level

O(n)

Understanding Time Complexity

We want to understand how the time to work with trees changes as the tree grows.

How does the size and shape of a tree affect how long operations take?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Simple function to find the height of a binary tree
func height(node *Node) int {
  if node == nil {
    return 0
  }
  leftHeight := height(node.Left)
  rightHeight := height(node.Right)
  if leftHeight > rightHeight {
    return leftHeight + 1
  }
  return rightHeight + 1
}
    

This code finds the height of a 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 tree grows, the function visits every node once to find height.

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

Pattern observation: Operations grow directly with the number of nodes.

Final Time Complexity

Time Complexity: O(n)

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

Common Mistake

[X] Wrong: "The height function only looks at the longest path, so it runs in constant time."

[OK] Correct: Even though height depends on the longest path, the function must visit every node to be sure, so time grows with total nodes.

Interview Connect

Understanding how tree operations scale helps you explain your approach clearly and confidently in interviews.

Self-Check

"What if the tree was balanced versus very unbalanced? How would that affect the time complexity of finding height?"

Practice

1. In a tree data structure, which node is called the root?

easy

A. The topmost node with no parent

B. A node with no children

C. Any node at the lowest level

D. A node with exactly two children

Solution

1. Step 1: Understand the root node definition

   The root node is the starting point of a tree and has no parent node above it.

2. Step 2: Differentiate root from other nodes

   Leaf nodes have no children, but root specifically has no parent. Lowest level nodes are leaves or internal nodes, not necessarily root.

3. Final Answer:

   The topmost node with no parent -> Option A

4. Quick Check:

   Root = topmost node with no parent [OK]

Hint: Root is always the top node without a parent [OK]

Common Mistakes:

• Confusing root with leaf node
• Thinking root must have children
• Assuming root is at the lowest level

2. Which of the following is the correct way to declare a tree node struct in Go with integer data and pointers to left and right children?

easy

A. type Node struct { data int, left &Node, right &Node }

B. type Node struct { data int, left *Node, right *Node }

C. type Node struct { data int, left Node, right Node }

D. type Node struct { data int, left Node*, right Node* }

Solution

1. Step 1: Recall Go pointer syntax

   In Go, pointers are declared with * before the type, e.g., *Node.

2. Step 2: Check each option's pointer usage

   type Node struct { data int, left *Node, right *Node } uses *Node correctly. type Node struct { data int, left Node, right Node } uses Node (value, not pointer). type Node struct { data int, left &Node, right &Node } uses &Node which is invalid syntax. type Node struct { data int, left Node*, right Node* } uses Node* which is invalid in Go.

3. Final Answer:

   type Node struct { data int, left *Node, right *Node } -> Option B

4. Quick Check:

   Pointer syntax in Go = *Type [OK]

Hint: Use *Type for pointers in Go structs [OK]

Common Mistakes:

• Using value type instead of pointer for children
• Using &Type or Type* which are invalid in Go
• Using semicolons instead of commas in struct fields

3. Given the following tree structure in Go, what is the depth of node with value 5?

type Node struct {
  data  int
  left  *Node
  right *Node
}

root := &Node{data: 1}
root.left = &Node{data: 2}
root.right = &Node{data: 3}
root.left.left = &Node{data: 4}
root.left.right = &Node{data: 5}

medium

A. 2

B. 1

C. 3

D. 4

Solution

1. Step 1: Understand depth definition

   Depth of a node is the number of edges from the root to that node.

2. Step 2: Trace path from root to node 5

   Path: 1 (root) -> 2 -> 5. Number of edges = 2.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Depth = edges from root = 2 [OK]

Hint: Count edges from root to node for depth [OK]

Common Mistakes:

• Counting nodes instead of edges
• Confusing depth with height
• Miscounting path length

4. Consider this Go code snippet for a binary tree node:

type Node struct {
  data  int
  left  *Node
  right *Node
}

func isLeaf(n *Node) bool {
  return n.left == nil && n.right == nil
}

root := &Node{data: 10}
root.left = &Node{data: 20}
root.right = &Node{data: 30}
root.left.left = &Node{data: 40}

fmt.Println(isLeaf(root.left))

What is the output and why?

medium

A. true

B. Runtime panic

C. Compilation error

D. false

Solution

1. Step 1: Understand isLeaf function

   It returns true if both left and right children are nil (no children).

2. Step 2: Check root.left node children

   root.left has left child (40) but no right child, so left != nil, right == nil.

3. Step 3: Evaluate isLeaf(root.left)

   Since left is not nil, isLeaf returns false.

4. Final Answer:

   false -> Option D

5. Quick Check:

   Leaf = no children, root.left has left child [OK]

Hint: Leaf nodes have no children; check both left and right nil [OK]

Common Mistakes:

• Assuming node with one child is leaf
• Confusing nil checks causing panic
• Expecting true without checking both children

5. You have a binary tree where the root is at level 0. If a node is at level 3, what is its height assuming the tree height is 5? Also, which node is at level 0? Choose the correct pair of answers.

hard

A. Height = 5; Root node is at level 0

B. Height = 3; Leaf node is at level 0

C. Height = 2; Root node is at level 0

D. Height = 2; Leaf node is at level 0

Solution

1. Step 1: Calculate height of node at level 3

   Height = total tree height - node level = 5 - 3 = 2.

2. Step 2: Identify node at level 0

   By definition, root node is at level 0.

3. Final Answer:

   Height = 2; Root node is at level 0 -> Option C

4. Quick Check:

   Height = 5 - 3 = 2; Root at level 0 [OK]

Hint: Height = tree height - level; root always level 0 [OK]

Common Mistakes:

• Confusing height with level
• Assuming leaf node at level 0
• Calculating height as level number