Question 3 (Caley Changes, 40 points). In our proof of Caley's Theorem, we discovered a bijection between labelled trees on n vertices and lists of n 2 numbers from 1 ton. We will see what happens when we make small changes to this list of numbers. (a) Let L be a list of n2 numbers between 1 and n. Let L' be obtained from L by swapping two adjacent numbers. Let T and T' be the trees corresponding to L and L'. Show that T and T' have at most three edges that are different (where two edges are considered the same if they connect endpoints with the same labels). [20 points] (b) Let L be a list of n 2 numbers between 1 and n. Let L' be obtained from L by changing the value of one number in the list. Let T and T' be the trees corresponding to L and L'. Show that for the correct choice of L and L', it is possible forT and T' to have as many as n1 many different edges. [20 points]