What are common examiner traps in Binary Heaps?

Confusing 0-based indexing (2i+1, 2i+2) with 1-based indexing (2i, 2i+1). Assuming finding the maximum element in a min-heap takes O(log n) instead of (n) across leaves. Off-by-one errors when counting leaves: forgetting that ⌊ n/2 ⌋ gives internal nodes, so leaf count is n - ⌊ n/2 ⌋ = ⌈ n/2 ⌉. Simulating Build-Heap from index 1 to n (top-down) instead of bottom-up from ⌊ n/2 ⌋ down to 1. Confusing height in terms of edges (⌊ log2 n ⌋) with height in terms of levels/nodes (⌊ log2 n ⌋ + 1). Violating the complete binary tree shape property on the last level while verifying max-heap condition.

GATE CSE Binary Heaps PYQs

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Binary heaps are tested frequently in GATE CS with a strong focus on structural and array properties (complete binary tree indexing), operation trace (insertion… Read the guide

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Questions: 15
Asked: since 2011
Total marks: 22
Marks / paper: 0.8

15 GATE CSE Binary Heaps PYQs with solutions, from Programming and Data Structures. Browse all GATE CSE PYQs.

Questions

Every official paper that asked this topic, newest first.

What GATE tests in Binary Heaps

Concepts, patterns, formulas and traps distilled from past papers.

Binary heaps are tested frequently in GATE CS with a strong focus on structural and array properties (complete binary tree indexing), operation trace (insertion bubbling up and bottom-up build-heap sift-down), and extreme value locations (finding maximum in min-heap or minimum in max-heap). Questions vary from direct index/leaf counting formulas and tree height calculations to asymptotic complexities of priority queue operations. Recent years exhibit an increased prevalence of NAT and MSQ questions testing index ranges and structural properties.

Core concepts

Complete binary tree shape property and array mapping
Max-heap and min-heap order properties
Bottom-up Build-Heap algorithm (starting from index ⌊ n/2 ⌋ down to 1)
Insertion and bubble-up (sift-up / percolate-up) mechanics
Extract-Max / Extract-Min and sift-down mechanics
Arbitrary node deletion `delete(i)` taking O(log n) / O(d) time
Properties and locations of leaf nodes vs internal nodes
Search complexity for non-root extreme elements (e.g., maximum in min-heap is (n))

Recurring question patterns

Leaf Node Indexing & Extreme Value Location
Commonmixed2 marks2026, 2024, 2021, 2020
Questions asking for the location, number of possible indices, or number of comparisons/complexity required to locate the opposite extreme element (e.g., maximum in a min-heap), relying on the fact that leaves occupy indices $\lfloor n/2 \rfloor + 1$ to $n$ .
Heap Operations Tracing & Simulation
CommonMCQ1 marks2024, 2015, 2014
Tracing step-by-step element insertions with percolate-up (bubble-up) or running bottom-up heap construction (Build-Heap) and reporting the resulting array elements or total swap/interchange count.
Heap Structural Validity & Identification
CommonMCQ1 marks2023, 2011
Identifying whether a given tree diagram or 1-D array represents a valid max/min heap by simultaneously verifying the complete binary tree structure and the parent-child heap order inequality ( $A[\lfloor i/2 \rfloor] \ge A[i]$ or $A[\lfloor i/2 \rfloor] \le A[i]$ ).
Heap Complexity & Height Properties
Commonmixed1.5 marks2025, 2023, 2016, 2015
Determining the height $\lfloor \log_2 n \rfloor$ , worst-case time complexities of operations (EXTRACT-MAX, INSERT, DELETE(i)), or single-node sift-down lower bounds.
Key Ancestry and Depth Constraints
RareNAT2 marks2016
Finding the maximum or minimum possible depth of a specific key in a min/max heap based on the number of strictly smaller/greater available ancestors.

Formulas GATE uses

Array Parent-Child Index Relations (1-based)

\text{Parent}(i) = \lfloor i/2 \rfloor, \quad \text{Left}(i) = 2i, \quad \text{Right}(i) = 2i+1

Used to verify heap properties in array representation or navigate the tree.

Leaf vs Internal Node Index Range (1-based)

\text{Internal indices: } 1 \le i \le \lfloor n/2 \rfloor, \quad \text{Leaf indices: } \lfloor n/2 \rfloor + 1 \le i \le n

Used when determining the number of leaf nodes $\lceil n/2 \rceil$ , identifying non-leaf indices, or locating maximum elements in a min-heap.

Height of a Binary Heap with $n$ Elements

h = \lfloor \log_2 n \rfloor

Used when computing the maximum number of edges from the root to any leaf node.

Max Comparisons Among $k$ Leaf Candidates

\text{Comparisons} = k - 1 = \lceil n/2 \rceil - 1

Used to compute minimum comparisons required to find the maximum in a min-heap or minimum in a max-heap.

Max Depth of Key $k$ in Min-Heap (keys $1 \dots n$)

\text{Max Depth}(k) = \min(k - 1, \lfloor \log_2 n \rfloor)

Used when finding the deepest possible placement of a key by allocating smaller keys along the ancestor path.

Examiner traps in Binary Heaps

Confusing 0-based indexing (2i+1, 2i+2) with 1-based indexing (2i, 2i+1).
Assuming finding the maximum element in a min-heap takes O(log n) instead of (n) across leaves.
Off-by-one errors when counting leaves: forgetting that ⌊ n/2 ⌋ gives internal nodes, so leaf count is n - ⌊ n/2 ⌋ = ⌈ n/2 ⌉.
Simulating Build-Heap from index 1 to n (top-down) instead of bottom-up from ⌊ n/2 ⌋ down to 1.
Confusing height in terms of edges (⌊ log2 n ⌋) with height in terms of levels/nodes (⌊ log2 n ⌋ + 1).
Violating the complete binary tree shape property on the last level while verifying max-heap condition.

How questions are asked

Level-order array identification after insertions or build-heap
Counting swaps/interchanges during bottom-up heap construction
Identifying valid max/min heap among multiple array sequences or tree diagrams
Calculating tree height or maximum node depth given n elements (NAT)
Evaluating candidate leaf or internal index expressions (MSQ)
Number of comparisons or asymptotic runtime for heap search/delete operations

Paper trends

Shift from purely visual tree and array validation MCQs to arithmetic/formulaic NAT and MSQ questions on array indices and leaf calculations.
2026, 2025, 2024, 2020
Repeated testing of the opposite extreme element in a heap (maximum in min-heap) in various formats: comparison count NAT (2020), time complexity MCQ (2021), and index count MCQ (2024).
2024, 2021, 2020
Sustained inclusion of 1-mark mechanical array tracing questions (build-heap and successive insertion).
2024, 2015, 2014

Difficulty notes

Easy: Direct verification of heap property on a small array/tree, computing heap height $\lfloor \log_2 n \rfloor$ , standard asymptotic complexity of INSERT/EXTRACT-MAX ( $O(\log n)$ ), or single-element bubble-up trace. Medium: Counting exact swaps during multi-step build-heap, calculating comparisons required to find max in min-heap among all leaves, calculating max possible depth for a given integer key, or finding the resulting array prefix after multiple downward siftings.