  We study minimal vertex covers of trees. Contrarily to the number $N_{vc}(A)$
of minimal vertex covers of the tree $A$, $\log N_{vc}(A)$ is a self-averaging
quantity. We show that, for large sizes $n$, $\lim_{n\to +\infty} <\log
N_{vc}(A)>_n/n= 0.1033252\pm 10^{-7}$. 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 $<\log N_{vc}>$ over totally
degenerate trees of given size, which we perform numerically.
