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Chapter 7: Problem 29

Show that a heap with \(n\) nodes has \(\lceil n / 2\rceil\) leaves.

Short Answer

Expert verified
The statement is true. A heap with \(n\) nodes indeed has \(\lceil n / 2 \rceil\) leaves. This is substantiated by understanding the structure of a heap. Roughly half of the nodes are leaves. The leaves in a binary heap start from the middle node and extend to the last node if the heap is represented as an array.

Step by step solution

01

Understand the structure of a heap

In a binary heap, each node has at most two children. The 'heapiness' property defines an order among the nodes based on their values. In a max heap, for instance, each parent node has a value greater than or equal to its children. Conversely, in a min heap, each parent node has a value lesser than or equal to its children. The crucial point here is that nearly half of the nodes in a binary heap are leaf nodes.
02

Relation between nodes and leaves

In a binary heap, any node from \(\lceil n/2 \rceil\) + 1 to n (n being the total number of nodes), is a leaf node by structure of binary heap, by which leaves start from middle to the end of heap if heap is represented in the form of an array.
03

Calculating the number of leaves

On counting these leaf nodes, we get the total number of leads as \(n-(\lceil n/2 \rceil)+1\). This simplifies to \(\lceil n/2 \rceil\), thus proving the initial proposition.

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Most popular questions from this chapter

Another way to sort a list by exchanging out-of-order keys is called Bubble Sort. Bubble Sort scans adjacent pairs of records and exchanges those found to have out-of-order keys, After the first time through the list, the record with the largest key (or the smallest key) is moved to its proper position. This process is done repeatedly on the remaining, unsorted part of the list until the list is completely sorted. Write the Bubble Sort algorithm. Analyze your algorithm, and show the results using order notation. Compare the performance of the Bubble Sort algorithm against those of Insertion Sort, Exchange Sort, and Selection Sort.

Use the divide-and-conquer approach to write a nonrecursive Mergesort algorithm. Analyze your algorithm, and show the results using order notation. Note that it will be necessary to explicitly maintain a stack in your algorithm.

Write a linear-time sorting algorithm that sorts a list of values of a given ordinal type

Write a nonrecursive Quicksort algorithm. Analyze your algorithm, and show the results using order notation. Note that it will be necessary to explicitly maintain a stack in your algorithm.

Implement the Quicksort algorithm using different strategies for choosing a pivot item, run it on your system, and study its best-case, average-case, and worst-case performances for different strategies using several problem instances.
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