Given a global field K and a rational function f(x) defined over K, one may take pre-images of 0 under successive iterates of f, and thus obtain an infinite tree by assigning edges according to the action of f. The absolute Galois group of K acts on the tree, giving a subgroup of the group of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. The analogy here is to Serre's finite index results for Galois representations arising from elliptic curves. I will discuss the contributions of several researchers, including Boston and Jones, along with my own work (joint with Jones) on these questions.