# pset7 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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EECS 310, Fall 2011 Instructor: Nicole Immorlica Problem Set #7 Due: November 15, 2011 1. (20 points) A k -ary tree is a tree in which each internal node has at most k children. The height h of a node is the length of the path from the root to the node. The height H of a tree is the maximum height of any node. (a) (10 points) Derive a formula for the maximum number of nodes of height h in a k -ary tree. Prove that your formula is correct using induction. (b) (10 points) Using the above, write a summation to compute the maximum number of nodes in a ternary tree of height at most H and compute the closed form. 2. (20 points) Consider an n × n grid. (a) (5 points) How many squares does the grid contain? (b) (5 points) How many rectangles does the grid contain? (c) (10 points) Suppose the grid is missing an (( n - k ) × ( n - k ))-sized piece from each of its corners (so it now looks like a cross in which each board is n × k ). How many squares does it contain now? 3. (20 points) Give combinatorial proofs of the following identities.

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pset7 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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