### **Homework 7, Fundamental Algorithms, Fall 97**

Due Date: Monday, October 27.

1. Consider a tree in which each internal node has c or more children,
where c is a constant integer of value 2 or more.
Show that the number of internal nodes, I, in the tree and the number
of leaves, L, are related by the formula: L-1 >= (c-1)I.

Hint: Use tokens. How many tokens should a leaf receive, and what do you
do with the tokens?

2. Consider a tree in which each node is colored either red or green.
You are allowed to cut edges in this tree.
Your goal is to choose the subset of edges to cut so that the resulting
subtree rooted at the original root r has the largest possible excess
of red over green nodes (possibly, this excess is negative).
Design an algorithm to compute this excess.

Hint. Compute this for every node in the tree.
How do you decide whether to cut an edge?

Problems 2.12, 4.16. (See back of text, pages 399-405).

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cole@cs.nyu.edu (Richard Cole)

Last modified: Oct 19, 1997