  We study the Tutte polynomial of two infinite families of finite graphs.
These are the Schreier graphs associated with the action of two well-known
self-similar groups acting on the binary rooted tree by automorphisms: the
first Grigorchuk group of intermediate growth, and the iterated monodromy group
of the complex polynomial $z^2-1$ known as the Basilica group. For both of
them, we describe the Tutte polynomial and we compute several special
evaluations of it, giving further information about the combinatorial structure
of these graphs.
