1. Using the tree implementation of disjoint sets answer the following questions. Proofs are not necessary for this question. (a) Describe a sequence of make-set and union operations that builds a tree such that the longest path has length 3 (3 edges on the path). Use the fewest number of elements possible. Also draw the resulting tree. (b) In general, what is the minimum number of nodes needed for the longest path length to be k? (c) What is the maximum number of nodes possible in a tree for the longest path length to be k?