I'm reading Ghrist's paper "<a href="http://www.math.upenn.edu/~ghrist/preprints/birman.pdf">Configuration spaces and braid groups on graphs in robotics</a>".
In <a href="https://mathoverflow.net/questions/139598/a-sufficient-condition-for-a-space-to-be-an-eilenberg-maclane-space">this</a> discussion, counterexamples are shown for both Theorem 2.3 and the implication "Theorem 2.3 $\Rightarrow$ Corollary 2.4". Are there other proofs, in literature, that configuration spaces of trees are Eilenberg-MacLane spaces? Or, is there a way to "fix" Ghrist's proof?