Pattern closure of groups of tree automorphisms

It is shown that every group of automorphisms of a regular rooted tree that is defined by forbidding a set of patterns of size s + 1 is the topological closure of a self-similar, countable, regular branch group, branching over its level s stabilizer. As an application, it is shown that there are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size two.