Data Structures Fundamentals

A hash table (or hash map) is the data structure that implements a dictionary, storing key-value pairs with efficient O(1) average-case lookup, insertion, and deletion. The term 'hash' alone refers to the hashing function or technique, not the structure. Trees and linked lists are fundamental structures but don't provide dictionary-like key-value storage.

When using a hash (implemented as an array), the parent of the entry at index i is found using integer division: floor(i/2). For example, index 6's parent is at index 3 (6/2=3), and index 5's parent is also at index 2 (5/2=2.5→2). Decimal division would give incorrect fractional results, while multiplication would give child indices, not the parent.

A stack follows LIFO (Last In First Out) principle. When you push t, then a, then p onto an empty stack, p is on top. Popping removes p first, then a, then t, giving the order p,a,t. This is the reverse of the push order.

A queue follows FIFO (First In First Out) principle. When you enqueue m, then a, then r into an empty queue, m is at the front. Dequeue removes m first, then a, then r, maintaining the original order m,a,r. This is different from a stack.

Graphs are ideal for representing networks where entities are connected by edges. Roads connecting cities is a classic graph application - cities are vertices and roads are edges. GUIs are hierarchical (tree-like), algebraic problems involve equations, and word-definition relationships are also graph-like but roads are the most direct example.

In a balanced binary tree, the maximum number of comparisons needed equals the tree height, which is log2(n) rounded up. For 15 elements, log2(15) ≈ 3.9, so 4 steps is the maximum. Option C (15) describes linear search, not binary search in a tree.

A stack is ideal for matching parentheses because it uses Last-In-First-Out (LIFO) order. When you see a closing parenthesis, you check if it matches the most recent opening one on the stack. This is the standard algorithm for expression validation.

A set is defined as a collection that cannot contain duplicate elements - this is the fundamental property that distinguishes sets from lists. Sets can store any type of data, not just numbers, and they do not maintain insertion order.

B+ Trees are preferred over B-Trees for filesystems because all data is stored in leaf nodes, making range queries and sequential access much faster. The leaf nodes are linked together, allowing efficient traversal of all records.