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Chapter 7: Problem 28

Show that a heap with \(n\) nodes has \([n / 2]\) leaves.

Short Answer

Expert verified
A heap with \(n\) nodes has \(\lfloor n / 2 \rfloor\) leaves

Step by step solution

01

Understand the structure of heap

In a complete binary tree (heap), levels are filled from left to right and till the last level, all levels are completely filled. Starting from the lowest level and going upwards, a binary tree has less or equal nodes with a higher level to it. Knowing this, a heap of \(n\) nodes can be divided into two sections. Half or more than half nodes are in the last level and the others are in all other levels combined. This is because every node except those at the last level, has 2 child nodes and every node at the last level is leaf node.
02

Prove the Theorem

Let's denote \(L\) as the number of leaf nodes and \(I\) as the number of internal nodes. From the definition of a heap, we know that every internal node has exactly 2 children, so \(2I = n - I\) because every node is either an internal node or the child of one. Solving this equation gives us \(I = n / 3\) and \(L = 2n / 3\), but since \(L\) must be an integer, we actually round down to get \(L = \lfloor n / 2 \rfloor\), proving the theorem.

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Most popular questions from this chapter

Write an algorithm that checks if an essentially complete binary tree is a heap. Analyze your algorithm and show the results using order notation.

Show that there are \(2^{j}\) nodes with depth \(j\) for \(j<\mathrm{d}\) in a heap having \(n\) (a power of 2 ) nodes. Here \(d\) is the depth of the heap.

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.

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 to those of Insertion Sort, Exchange Sort, and Selection Sort.

Write an algorithm that sorts a list of \(n\) elements in nonincreasing order by finding the largest and smallest elements and exchanges those elements with the elements in the first and last positions. Then the size of the list is reduced by \(2,\) excluding the two elements that are already in the proper positions, and the process is repeated on the remaining part of the list until the entire list is sorted. Analyze your algorithm and show the results using order notation.
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