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For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strandbraid group on $T$. We use discrete Morse theory techniques to show that thecohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicialcomplex $K_nT$ 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 $K_nT$.

Cohomology ring of tree braid groups and exterior face rings