  For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand
braid group on $T$. We use discrete Morse theory techniques to show that the
cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial
complex $K_nT$ that measures $n$-local interactions among essential vertices of
$T$. We show that, in many cases (for instance when $T$ is a binary tree),
$H^*(B_nT)$ is the exterior face ring determined by $K_nT$.
