The subgroup growth spectrum of virtually free groups

For a finitely generated group $\mathrm{\Gamma }$ denote by $\mu (\mathrm{\Gamma })$ the growth coefficient of $\mathrm{\Gamma }$, that is, the infimum over all real numbers $d$ such that $s_n(\mathrm{\Gamma })<n{!}^d$. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group. Moreover, we describe an algorithm to compute $\mu$.