If Γ is any finite graph, then the unlabelled configuration space of n points on Γ,
denoted UCnΓ, is the space of n–element subsets of Γ. The braid group of Γ on n
strands is the fundamental group of UCnΓ.
We apply a discrete version of Morse theory to these UCnΓ, for any n and any Γ,
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 UCnΓ strong
deformation retracts onto a CW complex of dimension at most k, where k is the
number of vertices in Γ of degree at least 3 (and k is thus independent of