  We explore the structure of the p-adic automorphism group Gamma of the
infinite rooted regular tree. We determine the asymptotic order of a typical
element, answering an old question of Turan.
  We initiate the study of a general dimension theory of groups acting on
rooted trees. We describe the relationship between dimension and other
properties of groups such as solvability, existence of dense free subgroups and
the normal subgroup structure. We show that subgroups of Gamma generated by
three random elements are full-dimensional and that there exist finitely
generated subgroups of arbitrary dimension. Specifically, our results solve an
open problem of Shalev and answer a question of Sidki.
