Chapter 17, Problem 3P

(a)

Program Plan Intro

To explain the rebuild the sub-tree rooted at node x so that it becomes 1/2-balanced.

(b)

Program Plan Intro

To explain performing a search in an n-node $\alpha$ balanced binary search tree take worse-case time of $O(\lg n)$ .

(c)

Program Plan Intro

To argue that any binary tree has nonnegative potential and that a 1/2 balanced tree has potential 0.

(d)

Program Plan Intro

To explain the value of c in terms of $\alpha$ in order for it to take $O(1)$ amortized time to rebuild a sub-tree that is not $\alpha$ balanced.

(e)

Program Plan Intro

To show that inserting a node into or deleting a node from an n -node $\alpha$ balanced tree cost $O(\lg n)$ amortized time.