Entire functions arising from trees

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function f with only two critical values ±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 the result of Grothendieck (see Schneps (1994)) to the case of infinite trees. Moreover, a similar idea leads to a new proof of the result of Nevanlinna (1932) and Elfving (1934)