We will talk about growth rates of subgroups of a free group. We prove that for every $\alpha \in [1,2r-1]$ there exists a subgroup $H < F_r$ whose growth rate is \alpha. We also prove a convergence result for the growth rate of the fundamental group of a graph. More precisely, we prove that if G is an (infinite) graph and $ \Gamma = \pi_1(G, v)$, then the growth rate of \Gamma equals the limit of the growth rate of $\Gamm_n := \pi_1(B_n, v)$, where $B_n$ is a ball of radius n around v.