In this paper we study growth functions of automatic and hyperbolic
groups. In addition to standard growth functions, we also want to
count the number of finite graphs isomorphic to a given finite graph
in the ball of radius $n$ around the identity element in the Cayley
graph. This topic was introduced to us by K. Saito [1991].
We report on fast methods to compute the growth function once we know
the automatic structure.
We prove that for a geodesic automatic structure, the growth function
for any fixed finite connected graph is a rational function.
For a word-hyperbolic group, we show that one can choose the denominator
of the rational function independently of the finite graph.