On the Minimum Size of a Contraction-Universal Tree

A tree T_uni is m-universal for the class of trees if for every tree T of size m, T can be obtained from T_uni by successive contractions of edges. We prove that a m-universal tree for the class of trees has at least mln(m) + (gamma-1)m + O(1) edges where is the Euler's constant and we build such a tree with less than mc edges for a fixed constant c = 1.984..