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MIT6_042JF10_assn05 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science October 5, 2010 Tom Leighton and Marten van Dijk Problem Set 5 Readings: Section 5.4 to 5.7 and 6.1-6.2. Problem 1. [20 points] Recall that a tree is a connected acyclic graph. In particular, a single vertex is a tree. We define a Splitting Binary Tree , or SBTree for short, as either the lone vertex, or a tree with the following properties: 1. exactly one node of degree 2 (called the root). 2. every other node is of degree 3 or 1 (called internal nodes and leaves, respectively). For the case of one single vertex (see above), that vertex is considered to be a leaf. It is easier to understand the definition visually, so an example is shown in Figure 1. An example of a tree which is not an SBTree is shown in Figure 2. (a) [10 pts] Show if an SBTree has more than one vertex, then the induced subgraph ob- tained by removing the unique root consists of two disconnected SBTrees. You may assume that by removing the root you obtain two separate connected componenents, so all you need to prove is that those two components are SBTrees. (b) [10 pts] Prove that two SBTrees with the same number of leaves must also have the same total number of nodes. Hint: As a conjecture, guess an expression for the total number of nodes in terms of the number of leaves N ( l ) . Then use induction to prove that it holds for all trees with the same l Problem 2. [20 points] In ”Die Hard: The Afterlife”, the ghosts of Bruce and Sam have been sent by the evil Simon on another mission to save midtown Manhattan. They have been told that there is a bomb on a street corner that lies in Midtown Manhattan, which Simon defines as extending from 41st Street to 59th Street and from 3rd Avenue to 9th Avenue. Additionally, the code that they need to defuse the bomb is on another street corner. Simon, in a good mood, also tosses them two carrots: He will have a helicopter initially lower them to the street corner where the bomb is.
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2 Problem Set 5 A E F D B C G Figure 1: Splitting Binary Tree: Node A is the root, B and E are internal
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