1. (Trees.) (a) A graph G on 2n vertices is said to have "doubled degrees" if it has ex actly 2 vertices with degree k, for every k{1,2,...n). For example, the graph drawn at right has doubled degrees How many trees exist with "doubled degrees?"Justify your claim, and find all such trees (b) In tutorial 7, we introduce the notion of a "rooted" tree, along with several related concepts (eg. height," "full m-ary tree.") Suppose that T is a rooted full binary tree on 99 vertices. What is the maximum height of T? What is the minimum height of T? Justify your claims (c) The wheel graph Wn is a graph on n + 1 vertices and 2n edges formed as follows: take a cycle graph Cn, and add a "center" vertex connected to every vertex in Cn Several examples are drawn at right Come up with a way of labeling the edges of Wn with the labels 1,1,2, 2,3,3,...n, n so that it contains exactly one minimal spanning tree Make sure that your labeling process is "genera .e. describe how to label Wn for any n, not ust Ws or W. Explain why your processes work. 18 marks