Complete vs full vs perfect binary trees in Data Structures Theory - Practice Questions

Challenge - 5 Problems

🧠 Conceptual

intermediate

Identify the type of binary tree

Which of the following best describes a full binary tree?

A A tree where every node has either 0 or 2 children.

B A tree where all levels except possibly the last are completely filled, and all nodes are as far left as possible.

C A tree where all internal nodes have two children and all leaves are at the same level.

D A tree where every node has at most one child.

💡 Hint

Think about the number of children each node can have in a full binary tree.

A full binary tree is defined as a tree where every node has either zero or two children. This means no node has only one child.

🧠 Conceptual

intermediate

Characteristics of a complete binary tree

Which statement correctly describes a complete binary tree?

A All levels except possibly the last are completely filled, and all nodes in the last level are as far left as possible.

B All levels are fully filled, and all leaves are at the same depth.

C Every node has either 0 or 2 children.

D Every node has at most one child.

💡 Hint

Consider how nodes are arranged especially on the last level.

A complete binary tree has all levels fully filled except possibly the last, which is filled from left to right without gaps.

🧠 Conceptual

advanced

Perfect binary tree properties

Which of the following is true about a perfect binary tree?

A It is a tree where all levels except the last are filled, and the last level is filled from left to right.

B It is a tree where every node has at most one child.

C It is a full binary tree where all leaves are at the same level, and all internal nodes have two children.

D It is a tree where nodes can have any number of children.

💡 Hint

Think about both fullness and leaf level uniformity.

A perfect binary tree is a full binary tree where all leaves are at the same depth, meaning the tree is completely balanced.

❓ Comparison

advanced

Comparing tree types by node count

Given a binary tree with height h, which tree type guarantees exactly 2^(h+1) - 1 nodes?

A Complete binary tree

B Full binary tree

C Any binary tree

D Perfect binary tree

💡 Hint

Recall the formula for the number of nodes in a perfectly balanced tree.

A perfect binary tree of height h has exactly 2^(h+1) - 1 nodes because all levels are fully filled.

❓ Reasoning

expert

Determining tree type from node arrangement

You have a binary tree where every level except the last is fully filled, the last level is filled from left to right, but some nodes have only one child. What type of binary tree is this?

A Full binary tree

B Complete binary tree

C Perfect binary tree

D Neither complete, full, nor perfect

💡 Hint

Focus on the arrangement of nodes and the presence of nodes with only one child.

A complete binary tree allows nodes to have one child on the last level as long as nodes are filled left to right. Full binary trees do not allow nodes with only one child, and perfect binary trees require all leaves at the same level.

Practice

1. Which of the following best describes a full binary tree?

easy

A. All levels are completely filled, including the last level.

B. Every node has either 0 or 2 children, no nodes have only one child.

C. All leaves are at the same level and every internal node has two children.

D. Nodes at the last level are as far right as possible.

Solution

1. Step 1: Understand the definition of a full binary tree

   A full binary tree is defined as a tree where every node has either zero or two children, meaning no node has only one child.

2. Step 2: Compare with other tree types

   Complete binary trees focus on filling levels left to right, and perfect binary trees are both full and complete with all leaves at the same level.

3. Final Answer:

   Every node has either 0 or 2 children, no nodes have only one child. -> Option B

4. Quick Check:

   Full binary tree = nodes have 0 or 2 children [OK]

Hint: Full means nodes have 0 or 2 children only [OK]

Common Mistakes:

• Confusing full with complete trees
• Thinking full means all levels filled
• Assuming nodes can have one child

2. Which property must a perfect binary tree always satisfy?

easy

A. All levels except the last are completely filled, and last level nodes are left aligned.

B. Nodes at the last level can be anywhere, not necessarily left aligned.

C. Every node has at most one child.

D. Every node has either 0 or 2 children, and all leaves are at the same level.

Solution

1. Step 1: Recall the definition of a perfect binary tree

   A perfect binary tree is both full and complete, meaning every node has 0 or 2 children and all leaves are at the same level.

2. Step 2: Eliminate incorrect options

   All levels except the last are completely filled, and last level nodes are left aligned. describes a complete tree, not necessarily perfect. Nodes at the last level can be anywhere, not necessarily left aligned. contradicts the left alignment rule. Every node has at most one child. is incorrect as perfect trees have nodes with two children.

3. Final Answer:

   Every node has either 0 or 2 children, and all leaves are at the same level. -> Option D

4. Quick Check:

   Perfect tree = full + all leaves same level [OK]

Hint: Perfect = full + all leaves at same level [OK]

Common Mistakes:

• Mixing complete and perfect tree definitions
• Ignoring leaf level uniformity
• Assuming last level nodes can be scattered

3. Consider the following binary tree structure:

       A
      / \
     B   C
    / 
   D 

Which type of binary tree is this?

medium

A. Complete binary tree

B. Full binary tree

C. Perfect binary tree

D. None of the above

Solution

1. Step 1: Analyze the tree structure

   The tree has root A with two children B and C. Node B has one child D. Node C has no children.

2. Step 2: Check properties against tree types

   Node B has exactly one child, so it is not full (full requires every node has 0 or 2 children). Levels 0 and 1 are completely filled; level 2 has D as far left as possible: complete. Leaves C (level 1) and D (level 2) not same level: not perfect.

3. Final Answer:

   Complete binary tree -> Option A

4. Quick Check:

   Last level left aligned but not full = complete tree [OK]

Hint: Complete trees fill left to right, full requires 0 or 2 children [OK]

Common Mistakes:

• Assuming missing right child means not complete
• Confusing full with complete
• Thinking perfect applies without full and complete

4. Identify the error in the following statement:
A perfect binary tree can have nodes with only one child.

medium

A. Incorrect, perfect trees require all internal nodes to have two children.

B. Correct statement, perfect trees allow one child nodes.

C. Incorrect, perfect trees only require last level to be full.

D. Correct, as long as the tree is complete.

Solution

1. Step 1: Understand perfect binary tree requirements

   Perfect binary trees are both full and complete, meaning every internal node must have exactly two children.

2. Step 2: Evaluate the statement

   The statement claims nodes can have only one child, which contradicts the full tree property required for perfect trees.

3. Final Answer:

   Incorrect, perfect trees require all internal nodes to have two children. -> Option A

4. Quick Check:

   Perfect tree = no single-child nodes [OK]

Hint: Perfect trees never have single-child nodes [OK]

Common Mistakes:

• Confusing complete with perfect tree rules
• Thinking one child allowed if tree is complete
• Ignoring full tree property in perfect trees

5. You have a binary tree with 15 nodes. It is known to be a perfect binary tree. How many leaf nodes does it have?

hard

A. 7

B. 16

C. 8

D. 15

Solution

1. Step 1: Recall leaf count formula for perfect binary trees

   In a perfect binary tree, the number of leaf nodes is (n + 1) / 2, where n is the total number of nodes.

2. Step 2: Calculate leaf nodes for 15 nodes

   Using the formula: (15 + 1) / 2 = 16 / 2 = 8 leaf nodes.

3. Final Answer:

   8 -> Option C

4. Quick Check:

   Leaf nodes = (total nodes + 1) / 2 = 8 [OK]

Hint: Perfect tree leaves = (nodes + 1) / 2 [OK]

Common Mistakes:

• Using total nodes as leaf count
• Confusing full and perfect tree leaf counts
• Forgetting leaf count formula