Height and depth of trees in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Depth of Nodes in a Tree

Consider a tree where the root node is at depth 0. What is the depth of a node that is three edges away from the root?

A 3

B 2

C 1

D 4

💡 Hint

Depth counts the number of edges from the root to the node.

The depth of a node is the number of edges from the root to that node. If a node is three edges away, its depth is 3.

📋 Factual

intermediate

Height of a Leaf Node

In a tree, what is the height of a leaf node?

A Depends on the tree

B 0

C 1

D Equal to its depth

💡 Hint

Height is the number of edges on the longest path from the node down to a leaf.

A leaf node has no children, so the longest path down from it is zero edges. Therefore, its height is 0.

🔍 Analysis

advanced

Calculating Tree Height from Node Heights

Given a tree where the root has three children with heights 2, 3, and 1 respectively, what is the height of the tree?

A 3

B 2

C 4

D 1

💡 Hint

The height of a node is 1 plus the maximum height among its children.

The root's height is 1 plus the maximum height of its children. Max child height is 3, so root height = 1 + 3 = 4. However, the question asks for the height of the tree, which is the height of the root node. So the correct answer is 4.

❓ Comparison

advanced

Difference Between Height and Depth

Which statement correctly describes the difference between height and depth of a node in a tree?

A Depth and height are always equal for any node.

B Height is the number of edges from the node to the root; depth is the number of edges from the node to the deepest leaf.

C Depth is the number of edges from the node to the root; height is the number of edges from the node to the deepest leaf.

D Depth counts nodes, height counts edges.

💡 Hint

Think about the direction each measure counts edges.

Depth measures distance from the root down to the node. Height measures distance from the node down to the farthest leaf. They are different concepts.

❓ Reasoning

expert

Determining Tree Height from Depths of Leaves

In a tree, the depths of all leaf nodes are 2, 3, 3, and 4. What is the height of the tree?

A 2

B 5

C 3

D 4

💡 Hint

The height of the tree equals the maximum depth among all leaves.

The height of the tree is the length of the longest path from the root to any leaf. Since the deepest leaf is at depth 4, the tree height is 4.

Practice

1. What does the depth of a node in a tree represent?

easy

A. The number of edges from the root to that node

B. The number of edges from that node to the farthest leaf

C. The total number of nodes in the tree

D. The number of children the node has

Solution

1. Step 1: Understand the definition of depth

   Depth is defined as the distance from the root node to the given node, measured in edges.

2. Step 2: Compare with other options

   Height measures distance to farthest leaf, not depth. Total nodes and children count are unrelated.

3. Final Answer:

   The number of edges from the root to that node -> Option A

4. Quick Check:

   Depth = edges from root to node [OK]

Hint: Depth counts edges from root down to the node [OK]

Common Mistakes:

• Confusing depth with height
• Thinking depth counts children
• Mixing depth with total nodes

2. Which of the following correctly describes the height of a leaf node in a tree?

easy

A. Height is always 1

B. Height is 0 because it has no children

C. Height equals the depth of the leaf

D. Height is the number of siblings it has

Solution

1. Step 1: Recall height definition for any node

   Height is the number of edges on the longest path from the node down to a leaf.

2. Step 2: Apply to leaf node

   A leaf node has no children, so the longest path down is zero edges, making height 0.

3. Final Answer:

   Height is 0 because it has no children -> Option B

4. Quick Check:

   Leaf height = 0 edges down [OK]

Hint: Leaf nodes always have height zero [OK]

Common Mistakes:

• Assuming height is 1 for leaves
• Confusing height with depth
• Counting siblings as height

3. Consider the following tree structure:

        A
       / \
      B   C
     /   / \
    D   E   F
           /
          G

What is the height of node C?

medium

A. 0

B. 1

C. 3

D. 2

Solution

1. Step 1: Identify the subtree rooted at node C

   Node C has children E and F; F has child G.

2. Step 2: Find longest path from C down to a leaf

   Paths: C->E (1 edge), C->F->G (2 edges). Longest path length is 2 edges.

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Height of C = longest path down = 2 edges [OK]

Hint: Height = longest edges down from node [OK]

Common Mistakes:

• Counting number of children instead of edges
• Confusing height with depth
• Ignoring deeper descendants

4. A student wrote that the depth of the root node in any tree is 1. What is wrong with this statement?

medium

A. Depth depends on number of children, not fixed

B. Depth of root is always equal to height

C. Depth of root is always 0, not 1

D. Depth cannot be defined for root node

Solution

1. Step 1: Recall definition of depth for root

   Depth is edges from root to node; root is at distance zero from itself.

2. Step 2: Identify error in student's statement

   Student incorrectly assigns depth 1 to root; correct depth is 0.

3. Final Answer:

   Depth of root is always 0, not 1 -> Option C

4. Quick Check:

   Root depth = 0 edges [OK]

Hint: Root node depth is zero by definition [OK]

Common Mistakes:

• Assigning depth 1 to root
• Confusing depth with height
• Thinking depth depends on children

5. Given a tree where the root node has depth 0 and height 4, and a node at depth 3 has height 1, what is the height of a leaf node at depth 4?

hard

A. 0

B. 1

C. 3

D. 4

Solution

1. Step 1: Understand height of leaf nodes

   Leaf nodes have height 0 because they have no children below.

2. Step 2: Apply to leaf at depth 4

   Since the node at depth 3 has height 1, its child at depth 4 must be a leaf with height 0.

3. Final Answer:

   0 -> Option A

4. Quick Check:

   Leaf node height = 0 [OK]

Hint: Leaf nodes always have height zero regardless of depth [OK]

Common Mistakes:

• Assuming height equals depth
• Thinking height increases with depth
• Confusing height with number of siblings