In this problem, we consider a decomposition of a tree that may be useful when

using a divide-and-conquer approach to solve some problem on that tree.

(a) Show that for any tree with n vertices, there is an edge such that removing it partitions

the tree into two subtrees each with at most 3n/4 vertices.

(b) Show that the constant in (a) is optimal. More precisely, show that there are arbitrarily

large trees for which the best partition factor is 3/4 (it is not sufficient to display a single

tree of some size).

(c) Show that given any tree with n vertices, by removing O(log n) edges, we can partition

the tree into two forests (collections of trees) A and B with numbers of vertices ⌈n/2⌉

and ⌊n/2⌋

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