Large arboreal Galois representations

Given a field K, a polynomial f∈K[x] of degree d, and a suitable element t∈K, the set of preimages of t under the iterates f^∘n carries a natural structure of a d-ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d≥20 is even and K=Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K=F(t) and f∈F[x] in which the corresponding Galois groups are the monodromy groups of the ramified covers f^∘n:P_F ¹→P_F ¹.