I'm reading Ghrist's paper "[Configuration spaces and braid groups on graphs in robotics](http://www.math.upenn.edu/~ghrist/preprints/birman.pdf)".
In [this](https://mathoverflow.net/questions/139598/a-sufficient-condition-for-a-space-to-be-an-eilenberg-maclane-space) 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?



Configuration spaces of trees are Eilenberg-MacLane spaces
The paper



> 
> Patty, C. W., *Homotopy groups of certain deleted product spaces,* 
>  Proc. Amer. Math. Soc. 12 (1961) 369–373. 
> 
> 
> 


contains a proof that for any connected, finite graph having essential vertices (of degree $\ge3$) the "deleted product" $$C\_2(\Gamma) = \Gamma\times\Gamma \setminus \Delta\Gamma$$ is aspherical. The proof uses the following Theorem of J.H.C. Whitehead:


Suppose $X=A\cup B$, where $X$, $A$ and $B$ are connected polyhedra. Suppose that $A\cap B$ has finitely many path components $C\_i$. If:


1. each of $A$, $B$ and $C\_i$ are aspherical, and
2. each map $\pi\_1(C\_i)\to \pi\_1(A)$ and $\pi\_1(C\_i)\to \pi\_1(B)$ induced by inclusion is injective,


then $X$ is aspherical.


I don't know if this proof can be adapted to give asphericity of $C\_N(\Gamma)$ for $N>2$ when $\Gamma$ is a tree, but perhaps this is what Ghrist had in mind.


