Minimal vertex covers of random trees

We study minimal vertex covers of trees. Contrarily to the number Nvc(A) of minimal vertex covers of the tree A, log Nvc(A) is a self-averaging quantity. We show that, for large sizes n, limn→+∞ < log Nvc(A) >n /n = 0.1033252 ± 10−7. The basic idea is, given a tree, to concentrate on its degenerate vertices, that is those vertices which belong to some minimal vertex cover but not to all of them. Deletion of the other vertices induces a forest of totally degenerate trees. We show that the problem reduces to the computation of the size distribution of this forest, which we perform analytically, and of the average < log Nvc > over totally degenerate trees of given size, which we perform numerically.