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Min-heap and max-heap properties in Data Structures Theory - Practice Problems & Coding Challenges

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🧠 Conceptual
intermediate
2:00remaining
Understanding Min-Heap Property

Which of the following best describes the min-heap property for a binary heap?

AEvery parent node is less than or equal to its children nodes.
BEvery parent node is greater than or equal to its children nodes.
CAll leaf nodes have smaller values than internal nodes.
DThe root node has the largest value in the heap.
Attempts:
2 left
💡 Hint

Think about which node should have the smallest value in a min-heap.

🧠 Conceptual
intermediate
2:00remaining
Max-Heap Root Value

In a max-heap, what can be said about the value of the root node compared to other nodes?

AIt is the smallest value in the heap.
BIt is always equal to one of the leaf nodes.
CIt is the largest value in the heap.
DIt is equal to the average of all node values.
Attempts:
2 left
💡 Hint

Consider the heap property that defines max-heaps.

🔍 Analysis
advanced
2:00remaining
Heap Property Violation Detection

Given the following array representing a binary heap: [10, 15, 20, 17, 25], which heap property does it violate if any?

Data Structures Theory
Array representation: [10, 15, 20, 17, 25]
ADoes not violate min-heap or max-heap properties.
BViolates max-heap property because 15 is less than 10.
CViolates min-heap property because 17 is less than 15.
DViolates min-heap property because 15 is greater than 10.
Attempts:
2 left
💡 Hint

Check if parents are smaller or larger than their children according to min-heap or max-heap rules.

❓ Comparison
advanced
2:00remaining
Comparing Min-Heap and Max-Heap Structures

Which statement correctly compares min-heaps and max-heaps?

AMax-heaps store elements in descending order in their array representation.
BBoth min-heaps and max-heaps are complete binary trees but differ in parent-child value relationships.
CMin-heaps always have smaller height than max-heaps for the same number of elements.
DMin-heaps allow duplicate values but max-heaps do not.
Attempts:
2 left
💡 Hint

Think about the shape and ordering rules of heaps.

❓ Reasoning
expert
2:00remaining
Effect of Inserting an Element in a Min-Heap

When inserting a new element into a min-heap, which of the following best describes the process to maintain the min-heap property?

AInsert the element at the root and swap downwards until the property is restored.
BInsert the element at the root and swap upwards until the property is restored.
CInsert the element at the last position and swap downwards until the property is restored.
DInsert the element at the last position and swap upwards until the property is restored.
Attempts:
2 left
💡 Hint

Consider where new elements are added and how to restore heap order.

Practice

(1/5)
1. Which of the following best describes a min-heap property?
easy
A. The tree is not necessarily complete.
B. The parent node is always larger than or equal to its children.
C. The root node is always the largest value.
D. The parent node is always smaller than or equal to its children.

Solution

  1. Step 1: Understand min-heap property

    A min-heap requires that every parent node is smaller than or equal to its children.

  2. Step 2: Compare options with definition

    The parent node is always smaller than or equal to its children. matches this definition exactly, while others describe max-heap or incorrect properties.

  3. Final Answer:

    The parent node is always smaller than or equal to its children. -> Option D

  4. Quick Check:

    Min-heap = parent ≤ children [OK]
Hint: Min-heap means smallest value at the top [OK]
Common Mistakes:
  • Confusing min-heap with max-heap property
  • Thinking the tree can be incomplete
  • Assuming root is largest in min-heap
2. Which of the following is the correct way to describe a max-heap?
easy
A. A binary tree where each parent is smaller than its children.
B. A complete binary tree where each parent is greater than or equal to its children.
C. A tree where the root is always the smallest value.
D. A binary tree that is not necessarily complete.

Solution

  1. Step 1: Recall max-heap definition

    A max-heap is a complete binary tree where each parent node is greater than or equal to its children.

  2. Step 2: Match options to definition

    A complete binary tree where each parent is greater than or equal to its children. correctly states this. Options A and C describe min-heap or incorrect properties, and D is false because heaps must be complete.

  3. Final Answer:

    A complete binary tree where each parent is greater than or equal to its children. -> Option B

  4. Quick Check:

    Max-heap = parent ≥ children [OK]
Hint: Max-heap means largest value at the root [OK]
Common Mistakes:
  • Mixing min-heap and max-heap definitions
  • Ignoring the completeness of the tree
  • Thinking root is smallest in max-heap
3. Given the max-heap array representation [50, 30, 40, 10, 20], what is the root value after inserting 60 and reheapifying?
medium
A. 60
B. 30
C. 40
D. 50

Solution

  1. Step 1: Insert 60 at the end of the heap

    Array becomes [50, 30, 40, 10, 20, 60].

  2. Step 2: Reheapify by comparing 60 with its parent

    60 is greater than 40 (its parent), so swap. New array: [50, 30, 60, 10, 20, 40]. Then compare 60 with 50 (new parent), swap again. Final array: [60, 30, 50, 10, 20, 40].

  3. Final Answer:

    60 -> Option A

  4. Quick Check:

    New root after insert = 60 [OK]
Hint: New max inserted bubbles up to root [OK]
Common Mistakes:
  • Not swapping inserted value up correctly
  • Confusing min-heap and max-heap behavior
  • Forgetting to reheapify after insertion
4. Identify the error in this min-heap array: [5, 3, 8, 10, 7].
medium
A. The root 5 is larger than child 3, violating min-heap property.
B. The tree is not complete.
C. The array is sorted incorrectly.
D. No error; this is a valid min-heap.

Solution

  1. Step 1: Check min-heap property for root and children

    Root is 5, left child is 3. Since 5 > 3, this violates the min-heap rule that parent ≤ children.

  2. Step 2: Verify completeness and sorting

    The tree is complete and array order is not required to be sorted, so no error there.

  3. Final Answer:

    The root 5 is larger than child 3, violating min-heap property. -> Option A

  4. Quick Check:

    Parent ≤ children violated at root [OK]
Hint: Parent must be ≤ children in min-heap [OK]
Common Mistakes:
  • Assuming array must be sorted
  • Ignoring parent-child value checks
  • Confusing completeness with heap property
5. You have a min-heap represented as [4, 10, 15, 20, 30]. You want to replace the root with 25 and restore the min-heap property. What will be the new root after reheapifying?
hard
A. 15
B. 4
C. 10
D. 20

Solution

  1. Step 1: Replace root with 25

    Array becomes [25, 10, 15, 20, 30].

  2. Step 2: Reheapify by pushing 25 down

    Compare 25 with children 10 and 15. The smallest child is 10, so swap 25 and 10. New array: [10, 25, 15, 20, 30].

  3. Step 3: Continue reheapify

    Now 25 is at index 1 with children 20 and 30. 25 ≤ 20 and 30 is false, so swap 25 with 20. New array: [10, 20, 15, 25, 30].

  4. Step 4: Check if heap property holds

    25 is now a leaf node, so heap property restored.

  5. Final Answer:

    10 -> Option C

  6. Quick Check:

    New root after replace and reheapify = 10 [OK]
Hint: Replace root, push down smaller child until heap restored [OK]
Common Mistakes:
  • Not pushing down the replaced root correctly
  • Swapping with larger child instead of smaller
  • Stopping reheapify too early