1. Suppose we have a set of dynamic intervals, whose endpoints are
integers in the range 1, 2, ..., n, subject to operations in which an
individual interval is inserted or deleted. At any point in this
process, we wish to find the value in the given range that belongs to
the largest possible number of intervals.  Describe how to augment the
nodes of a segment tree with additional information that would allow one
to quickly find this most heavily covered point, and that could be
quickly updated when an interval is inserted or deleted.

2. Describe a set of n intervals for which an interval tree would always
use a total of n nodes, no matter which binary tree it is based on.

3. Suppose that you are going to search for a query value q in a
sequence of sorted lists in which the lengths of the lists are 2^1, 2^2,
2^4, 2^8, 2^16, etc. Would it help speed up the search to perform
fractional cascading on these lists? Why or why not?