  We study the homology of an explicit finite-index subgroup of the
automorphism group of a partially commutative group, in the case when its
defining graph is a tree. More concretely, we give a lower bound on the first
Betti number of this subgroup, based on the number and degree of a certain type
of vertices, which we call deep. We then use combinatorial methods to analyze
the average value of this Betti number, in terms of the size of the defining
tree.
