Cohomology ring of tree braid groups and exterior face rings

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