  Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration
space $UC^n \Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element
subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted
$B_n\Gamma$, is the fundamental group of $UC^n \Gamma$.
  We use the methods and results of our paper "Discrete Morse theory and graph
braid groups" to get a partial description of the cohomology rings $H^*(B_n
T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a
right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a
large number of counterexamples to Ghrist's conjecture that braid groups of
planar graphs are right-angled Artin groups.
