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

Time Complexity: Tree Terminology Root Leaf Height Depth Level

O(n)

Understanding Time Complexity

When working with trees, it is important to understand how the time to visit nodes grows as the tree gets bigger.

We want to know how operations like finding height or depth take longer as the tree size increases.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function treeHeight(node: TreeNode | null): number {
  if (node === null) return -1;
  const leftHeight = treeHeight(node.left);
  const rightHeight = treeHeight(node.right);
  return 1 + Math.max(leftHeight, rightHeight);
}
    

This code calculates 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 the height.

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

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 is constant time."

[OK] Correct: Even though height is about the longest path, the function must check every node to be sure, so it visits all nodes.

Interview Connect

Understanding how tree operations scale helps you explain your approach clearly and shows you know how data size affects performance.

Self-Check

"What if we changed the tree to be a linked list (all nodes have only one child)? How would the time complexity change?"

Practice

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

easy

A. A node with no children

B. The topmost node with no parent

C. Any node at the bottom 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: Compare options with definition

   The topmost node with no parent matches the definition exactly. Other options describe leaf nodes or specific child counts.

3. Final Answer:

   The topmost node with no parent -> Option B

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 bottom

2. Which of the following is the correct TypeScript syntax to define a simple tree node interface with a value and children array?

easy

A. interface TreeNode { value: number; children: TreeNode[]; }

B. interface TreeNode { value: number; children: TreeNode; }

C. interface TreeNode { value: number; children: number[]; }

D. interface TreeNode { value: number; child: TreeNode[]; }

Solution

1. Step 1: Check correct property names and types

   The children property should be an array of TreeNode objects, so type is TreeNode[].

2. Step 2: Validate each option

   interface TreeNode { value: number; children: TreeNode[]; } correctly uses 'children: TreeNode[]'. interface TreeNode { value: number; children: TreeNode; } uses single TreeNode, not array. interface TreeNode { value: number; children: number[]; } uses number array incorrectly. interface TreeNode { value: number; child: TreeNode[]; } uses wrong property name 'child'.

3. Final Answer:

   interface TreeNode { value: number; children: TreeNode[]; } -> Option A

4. Quick Check:

   Children is array of TreeNode objects [OK]

Hint: Children property must be an array of TreeNode [OK]

Common Mistakes:

• Using singular 'child' instead of 'children'
• Setting children as single object, not array
• Using wrong type like number[] for children

3. Given the following tree structure in TypeScript, what is the height of the tree?

const tree = {
  value: 1,
  children: [
    { value: 2, children: [] },
    { value: 3, children: [
      { value: 4, children: [] }
    ] }
  ]
};

medium

A. 1

B. 4

C. 3

D. 2

Solution

1. Step 1: Understand height definition

   Height is the longest path from root to any leaf node, counting edges.

2. Step 2: Trace longest path in given tree

   Root (1) to node (3) to node (4) is longest path with 2 edges, so height = 2. Counting nodes gives 3 levels, but height counts edges, so height = 2.

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Height = longest edge count from root to leaf = 2 [OK]

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

Common Mistakes:

• Counting nodes instead of edges for height
• Confusing height with depth
• Ignoring deeper branches

4. Consider this TypeScript code snippet for calculating the depth of a node in a tree. Identify the bug:

function getDepth(node: TreeNode, root: TreeNode, depth = 0): number {
  if (node === root) return depth;
  return getDepth(node, root.children[0], depth + 1);
}

medium

A. It always checks only the first child, missing other branches

B. It never increments depth

C. It compares node and root incorrectly

D. It should return depth + 1 instead of depth

Solution

1. Step 1: Analyze recursive call logic

   The function always calls getDepth on root.children[0], ignoring other children, so it misses nodes not in first child's subtree.

2. Step 2: Understand impact on depth calculation

   This causes incorrect depth for nodes not in first child's branch, as they are never reached.

3. Final Answer:

   It always checks only the first child, missing other branches -> Option A

4. Quick Check:

   Recursion must explore all children, not just first [OK]

Hint: Check if recursion covers all children, not just one [OK]

Common Mistakes:

• Assuming single child covers whole tree
• Not handling multiple branches in recursion
• Confusing depth increment logic

5. You have a tree where each node has a level property indicating its distance from the root (root level = 0). If a node has level 3, what is its depth and height relative to the root?

hard

A. Depth = 0, Height = 3

B. Depth = 3, Height = 3

C. Depth = 3, Height depends on subtree size

D. Depth and Height both equal to 0

Solution

1. Step 1: Understand level and depth relation

   Level and depth both measure distance from root, so depth = level = 3.

2. Step 2: Understand height meaning

   Height is longest path from node down to leaf, which depends on subtree size, so height is unknown from level alone.

3. Final Answer:

   Depth = 3, Height depends on subtree size -> Option C

4. Quick Check:

   Depth = level; height varies by subtree [OK]

Hint: Level = depth; height depends on node's subtree [OK]

Common Mistakes:

• Confusing height with depth or level
• Assuming height equals level always
• Ignoring subtree size for height