  Given any infinite tree in the plane satisfying certain topological
conditions, we construct an entire function $f$ with only two critical values
$\pm 1$ and no asymptotic values such that $f^{-1}([-1,1])$ is ambiently
homeomorphic to the given tree. This can be viewed as a generalization of a
result of Grothendieck to the case of infinite trees. Moreover, a similar idea
leads to a new proof of a result of Nevanlinna and Elfving.
