1. Show the following regarding the maximum item in the heap:

a. It must be at one of the leaves.

b. There are exactly fN/21 leaves.

c. Every leaf must be examined to ?nd it.

2. Show that the expected depth of the kth smallest element in a large complete heap (you may assume N  = 2k  - 1) is bounded by log k.

3. a. Give an algorithm to ?nd all nodes less than some value, X, in a binary heap. Your algorithm should run in O(K), where K is the number of nodes output.

b. Does your algorithm extend to any of the other heap structures discussed in this chapter?

c.  Give an algorithm that ?nds an arbitrary item X in a binary heap using at most roughly 3N/4 comparisons.