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 . 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.
"Growth functions and automatic groups." Experiment. Math. 5 (4) 297 - 315, 1996.