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 such that f−1([−1, 1]) is ambiently homeomorphic to given tree. This can be viewed as a generalization of result Grothendieck (see Schneps (1994)) case trees. Moreover, similar idea leads new proof Nevanlinna (1932) Elfving (1934).