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### Life Cycle Simulation Experiments Suppose that you want to simulate the life cycle of a Binary Search Tree

INSTRUCTIONS TO CANDIDATES

Life Cycle Simulation Experiments

Suppose that you want to simulate the life cycle of a Binary Search Tree as described on slides 45-48 of the Trees lecture slides (CIS 256 and CIS 279 Class Notes) and in section 4.3.5. (Average-Case Analysis) of the Textbook by Mark Allen Weiss.

Considering the problems that can be caused by random insert/remove pairs, here is a strategy that is not perfectly random, but close enough:

Initial Setup: Build a tree with N elements by inserting N elements chosen at random from the range 1 to M , where M is an integral multiple of N, i.e. M = K*N (where K is an integer independent of N).

Evolution of Binary Search Tree: Perform N2 pairs of insertions followed by deletions. Assume the existence of a routine, randomInteger(a, b), which returns a uniform random integer between a and b inclusive.

1. Briefly explain how to generate a random integer between 1 and M that is not already in the tree (so a random insertion can be performed). In terms of N and K, what is the probability of finding a new element to insert. (Your answer should not exceed half a typed page.)
2. Explain how to generate a random integer between 1 and M that is already in the tree (so that a random deletion of that integer can be performed). In terms of N and K, what is the probability of finding a new element to delete.  (Your answer should not exceed another half a typed page.)

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