Binary search trees with equal keys. This is problem 12-1 in the book. Equal keys pose a problem for the implementation of binary search trees. a. What is the asymptotic performance of Tree-Insert when used to insert n items with identical keys into an initially empty binary search tree? We propose to improve Tree-Insert by testing before line 5 to determine whether z:key == x:key and by testing before line 11 to determine whether z:key == y:key. If equality holds, we implement one of the following strategies. For each strategy, find the asymptotic performance of inserting n items with identical keys into an initially empty binary search tree. (The strategies are described for line 5, in which we compare the keys of z and x. Substitute y for x to arrive at the strategies for line 11.) b. Keep a boolean flag x:b at node x, and set x to either x:left or x:right based on the value of x:b, which alternates between false and true each time we visit x while inserting a node with the same key as x. c. Keep a list of nodes with equal keys at x, and insert z into the list. d. Randomly set x to either x:left or x:right. (Give the worst-case performance and informally derive the expected running time.) 1
