We study randomness in the automorphism group of the binary tree of depthn and its generalizations. These groups have an important role in grouptheory, and they also arise in connection with complex dynamics, fractalsand finite automata. We use branching processes to determine theasymptotic order of a random element, answering an old question of Turan.We show that three random elements generate a large subgroup with highprobability, leading to the solution of a problem of Shalev.This is joint work with M. Abert.