Question

Let \(T\) be a complete binary tree such that node \(v\) stores the key-entry pairs \((f(v), 0),\) where \(f(v)\) is the level number of \(v .\) Is tree \(T\) a heap? Why or why not?

Short answer

Tree \(T\) is a min-heap because the key of each node (its level number) is less than or equal to the keys of its children.

Step-by-step solution

Understand the Definitions

A complete binary tree is a binary tree where all levels are completely filled except possibly for the last level, which is filled from left to right. A heap is a complete binary tree where every node's key is greater than or equal to the keys of its children (max-heap) or every node's key is less than or equal to the keys of its children (min-heap).

Identify the Key-Entry Pairs

Each node in the tree stores a pair \((f(v), 0)\), where \(f(v)\) is the level number of the node \(v\). This means that the key associated with each node is its level number in the tree. For example, the root would have \(f(v) = 0\), the nodes at the second level would have \(f(v) = 1\), and so on.

Verify the Heap Property

To determine if the tree is a heap, compare the key of each node with the keys of its children. In a max-heap, the key of each node should be greater than or equal to its children's keys. In a min-heap, the key of each node should be less than or equal to its children's keys. Since the root's key is 0, and nodes at the second level have keys of 1, the root does not satisfy the max-heap property (0 < 1) but satisfies min-heap property.

Generalize for All Levels

For the given tree, as we go deeper into the levels, the key values increase (0, 1, 2, ...). Thus, \(f(v)\) of any parent node \(u\) is always less than \(f(v)\) of its children. This means each parent node's key is less than its children's keys, satisfying the min-heap property but not the max-heap property.

Key concepts

Binary Tree

A binary tree is a hierarchical data structure in which each node has, at most, two children. These children are referred to as the left child and the right child. Binary trees are fundamental in computer science and are used in various applications. The concept of a binary tree is essential for understanding more advanced data structures like heaps and binary search trees (BSTs). Key properties of binary trees include:

• Root: The topmost node in the tree.
• Leaf Nodes: Nodes that do not have any children.
• Subtrees: The left and right children of a node each constitute a subtree.

In a complete binary tree, all levels are fully filled except possibly the last, which should be filled from left to right. This structure ensures balance and efficiency in operations like insertion and deletion.

Heap

A heap is a special type of binary tree with specific properties. It can be either a max-heap or a min-heap. The key feature of a heap is its unique ordering:

• Max-Heap: A binary tree where each parent node has a key greater than or equal to the keys of its children. The largest key is found at the root.
• Min-Heap: A binary tree where each parent node has a key less than or equal to the keys of its children. The smallest key is located at the root.

The heap property ensures that the tree is always balanced, which makes heaps efficient in implementing priority queues and for heap sort. Heaps are considered complete binary trees, ensuring minimal height and efficient operations.

Min-Heap

A min-heap is a type of heap where each parent node's key is less than or equal to the keys of its children. This property ensures that the minimum key is always at the root of the tree. In the context of the given exercise, analyzing a complete binary tree with nodes storing key-entry pairs \((f(v), 0)\) whereby \(f(v)\) represents the level number of node \(v\), we find:

• The key at the root (level 0) is 0.
• The keys of nodes at level 1 are 1, at level 2 are 2, and so on.
• Every parent node key \(f(v)\) is less than the keys of its children, maintaining the min-heap property.

Therefore, the tree described in the exercise is a min-heap as every node key is less than or equal to its children's keys.