  If Gamma is any finite graph, then the unlabelled configuration space of n
points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of
Gamma. The braid group of Gamma on n strands is the fundamental group of
UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma),
for any n and any Gamma, and provide a clear description of the critical cells
in every case. As a result, we can calculate a presentation for the braid group
of any tree, for any number of strands. We also give a simple proof of a
theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a
CW complex of dimension at most k, where k is the number of vertices in Gamma
of degree at least 3 (and k is thus independent of n).
