hw04solutions - CSE 331 Spring 2008, Section 1 Homework 4...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
CSE 331 Spring 2008, Section 1 Homework 4 Solutions 1. Let h = 0. This gives us a binary “tree” with a single node. For this case, 2 h +1 – 1 = 2 1 – 1 = 2 – 1 = 1. Thus, the maximum number of nodes in a binary tree of height h is 2 h +1 – 1 when h = 0. Assume that for some k ≥ 0, the maximum number of nodes in a binary tree of height k is known to be 2 k +1 – 1. Consider a binary tree of height k + 1. The root node can have at most two subtrees. Each subtree can have a height of at most k . By our inductive hypothesis, these subtrees can contain a maximum of 2 k +1 – 1 nodes. Considering these two subtrees and the root, the maximum number of nodes in a binary tree of height k + 1 is therefore 2 * (2 k +1 – 1) + 1. Distributing terms, 2 * (2 k +1 – 1) + 1 = 2 * 2 k +1 – 2 + 1 = 2 ( k +1) + 1 – 1. By induction, the maximum number of nodes in a binary tree of height h is 2 h +1 – 1 for any h ≥ 0. 2. (a) At depth d of a perfect binary tree, there will be 2 d nodes. The depths of the nodes will range from 0 to h , the height of the tree. In #1, we established that there are 2 h +1 – 1 nodes in a perfect binary tree of height h . Therefore, the average depth of the nodes will be d = 0 h d 2 d / 2 h 1 1 . To figure out what sort of values we can expect, it would be very nice to get rid of the summation. If we write out a few terms of the series starting from h = 1, we get 2, 10, 34, 98, 258, 642. .. If we subtract 2 from each of these terms, we get the much nicer-looking series 0, 8, 32, 96, 256, 640. .. Factoring each of these terms gets us the still-nicer-looking series 0, 2 3 , 2 5 , 3 * 2 5 , 2 8 , 5 * 2 7 ... We can then rearrange the terms slightly to get 0 * 2 2 , 1 * 2 3 , 2 * 2 4 , 3 * 2 5 , 4 * 2 6 , 5 * 2 7 . From this, it seems reasonable to guess that d = 0 h d 2 d = h 1 2 h 1 2 . We can prove this by induction. Let
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/25/2008 for the course CSE 331 taught by Professor M.mccullen during the Spring '08 term at Michigan State University.

Page1 / 5

hw04solutions - CSE 331 Spring 2008, Section 1 Homework 4...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online