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

Practice - 5 Tasks

1fill in blank

easy

Complete the sentence: A full binary tree is a tree where every node has either {{BLANK_1}} or zero children.

Data Structures Theory

A full binary tree is a tree where every node has either [1] or zero children.

Drag options to blanks, or click blank then click option'

Aone

Btwo

Cthree

Dfour

Attempts:

3 left

💡 Hint

Common Mistakes

Thinking a full binary tree can have nodes with only one child.

Confusing full binary tree with complete binary tree.

A full binary tree means each node has exactly two children or none at all.

2fill in blank

medium

Complete the sentence: A complete binary tree is a tree where all levels are fully filled except possibly the {{BLANK_1}} level, which is filled from left to right.

Data Structures Theory

A complete binary tree is a tree where all levels are fully filled except possibly the [1] level, which is filled from left to right.

Drag options to blanks, or click blank then click option'

Asecond

Bfirst

Cmiddle

Dlast

Attempts:

3 left

💡 Hint

Common Mistakes

Choosing the first or middle level instead of the last level.

Confusing complete binary tree with full binary tree.

In a complete binary tree, all levels except possibly the last are fully filled, and the last level is filled from left to right.

3fill in blank

hard

Fix the error in the sentence: A perfect binary tree is a tree where all internal nodes have two children and all leaves are at {{BLANK_1}} level.

Data Structures Theory

A perfect binary tree is a tree where all internal nodes have two children and all leaves are at [1] level.

Drag options to blanks, or click blank then click option'

Athe same

Bvarious

Cdifferent

Drandom

Attempts:

3 left

💡 Hint

Common Mistakes

Thinking leaves can be at different levels in a perfect binary tree.

Confusing perfect binary tree with complete binary tree.

In a perfect binary tree, all leaves are at the same level, making the tree completely balanced.

4fill in blank

hard

Fill both blanks to complete the definition: A {{BLANK_1}} binary tree is always a {{BLANK_2}} binary tree, but not all {{BLANK_2}} binary trees are {{BLANK_1}}.

Data Structures Theory

A [1] binary tree is always a [2] binary tree, but not all [2] binary trees are [1].

Drag options to blanks, or click blank then click option'

Aperfect

Bcomplete

Cfull

Dbalanced

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up full and complete trees in this relationship.

Assuming all complete trees are perfect.

A perfect binary tree is always full because all internal nodes have two children and all leaves are at the same level. However, a full binary tree does not have to be perfect.

5fill in blank

hard

Fill all four blanks to complete the statement: In a {{BLANK_1}} binary tree, nodes are filled level by level from left to right; in a {{BLANK_2}} binary tree, every node has {{BLANK_3}} or zero children; and in a {{BLANK_4}} binary tree, all leaves are at the same level.

Data Structures Theory

In a [1] binary tree, nodes are filled level by level from left to right; in a [2] binary tree, every node has [3] or zero children; and in a [4] binary tree, all leaves are at the same level.

Drag options to blanks, or click blank then click option'

Acomplete

Bfull

Ctwo

Dperfect

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing which tree type fills nodes level by level.

Mixing the number of children nodes have in full trees.

A complete binary tree fills nodes level by level from left to right. A full binary tree has nodes with either two or zero children. A perfect binary tree has 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