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( 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 ( 2 n ) with height in terms of levels/nodes ( 2 n + 1). Violating the complete binary tree shape property on the last level while verifying max-heap condition.

Binary Heaps

15 Qs · since 2011 · 22 marks · 0.8 marks/paperStandard yield

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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 bubbling up and bot… Guide

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GATE CSE 2026 Set 1Q23MSQ1 markEasyLet $n$ be an odd number greater than 100. Consider a binary minheap with $n$ elements stored in an array $P$ whose index starts from 1. Which of the following indices of $P$ do/does NOT correspond to any leaf node of the minheap?
GATE CSE 2025 Set 1Q35NAT1 markEasyThe height of any rooted tree is defined as the maximum number of edges in the path from the root node to any leaf node. Suppose a Min-Heap $T$ stores 32 keys. The height of $T$ is ____________. (Answer in integer)
GATE CSE 2024 Set 1Q41MCQ2 marksMediumAn array [82, 101, 90, 11, 111, 75, 33, 131, 44, 93] is heapified. Which one of the following options represents the first three elements in the heapified array?
GATE CSE 2024 Set 1Q43MCQ2 marksEasyConsider a binary min-heap containing 105 distinct elements. Let $k$ be the index (in the underlying array) of the maximum element stored in the heap. The number of possible values of $k$ is
GATE CSE 2023Q12MCQ1 markEasyWhich one of the following sequences when stored in an array at locations $A[1], \ldots, A[10]$ forms a max-heap?
GATE CSE 2023Q46MCQ2 marksEasyLet $A$ be a priority queue for maintaining a set of elements. Suppose $A$ is implemented using a max-heap data structure. The operation $\text{EXTRACT-MAX}(A)$ extracts and deletes the maximum element from $A$ . The operation $\text{INSERT}(A, key)$ inserts a new element $key$ in $A$ . The properties of a…
GATE CSE 2021 Set 2Q2MCQ1 markEasyLet $H$ be a binary min-heap consisting of $n$ elements implemented as an array. What is the worst case time complexity of an optimal algorithm to find the maximum element in $H$ ?
GATE CSE 2020Q47NAT2 marksMediumConsider the array representation of a binary min-heap containing 1023 elements. The minimum number of comparisons required to find the maximum in the heap is _________.
GATE CSE 2016 Set 1Q37MCQ2 marksMediumAn operator delete(i) for a binary heap data structure is to be designed to delete the item in the $i$ -th node. Assume that the heap is implemented in an array and $i$ refers to the $i$ -th index of the array. If the heap tree has depth $d$ (number of edges on the path from the root to the farthest…
GATE CSE 2016 Set 2Q34NAT2 marksMediumA complete binary min-heap is made by including each integer in $[1, 1023]$ exactly once. The depth of a node in the heap is the length of the path from the root of the heap to that node. Thus, the root is at depth 0. The maximum depth at which integer 9 can appear is _________.
GATE CSE 2015 Set 1Q32MCQ2 marksMediumDiagramConsider a max heap, represented by the array: 40, 30, 20, 10, 15, 16, 17, 8, 4. Now consider that a value 35 is inserted into this heap. After insertion, the new heap is
GATE CSE 2015 Set 2Q17MCQ1 markEasyConsider a complete binary tree where the left and the right subtrees of the root are max-heaps. The lower bound for the number of operations to convert the tree to a heap is
GATE CSE 2015 Set 3Q19MCQ1 markMediumConsider the following array of elements. $\langle 89, 19, 50, 17, 12, 15, 2, 5, 7, 11, 6, 9, 100 \rangle$ The minimum number of interchanges needed to convert it into a max-heap is
GATE CSE 2014 Set 2Q12MCQ1 markEasyA priority queue is implemented as a Max-Heap. Initially, it has 5 elements. The level-order traversal of the heap is: 10, 8, 5, 3, 2. Two new elements 1 and 7 are inserted into the heap in that order. The level-order traversal of the heap after the insertion of the elements is:
GATE CSE 2011Q23MCQ1 markEasyA max-heap is a heap where the value of each parent is greater than or equal to the value of its children. Which of the following is a max-heap?

Topic guide

What does GATE test in Binary Heaps?

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( 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
common · mixed · 2 marks · 2026, 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
common · MCQ · 1 marks · 2024, 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
common · MCQ · 1 marks · 2023, 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
common · mixed · 1.5 marks · 2025, 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
rare · NAT · 2 marks · 2016
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( 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 ( 2 n ) with height in terms of levels/nodes ( 2 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.

Binary Heaps

15 Qs · since 2011 · 22 marks · 0.8 marks/paperStandard yield

Take topic test

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 bot… Guide

Type

Topic guide

What does GATE test in Binary Heaps?

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( 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
common · mixed · 2 marks · 2026, 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
common · MCQ · 1 marks · 2024, 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
common · MCQ · 1 marks · 2023, 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
common · mixed · 1.5 marks · 2025, 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
rare · NAT · 2 marks · 2016
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( 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 ( 2 n ) with height in terms of levels/nodes ( 2 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