  A self-similar group of finite type is the profinite group of all
automorphisms of a regular rooted tree that locally around every vertex act as
elements of a given finite group of allowed actions. We provide criteria for
determining when a self-similar group of finite type is finite,
level-transitive, or topologically finitely generated. Using these criteria and
GAP computations we show that for the binary alphabet there is no infinite
topologically finitely generated self-similar group given by patterns of depth
$3$, and there are $32$ such groups for depth $4$.
