The $ k $-configuration space $ B_kΓ$ of a topological space $ Γ$ is the space of sets of $ k $ distinct points in $ Γ$. In this paper, we consider the case where $ Γ$ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_kΓ$ is given by a polynomial $P_Γ^i(k)$ in $ k $, called the Hilbert polynomial of $ Γ$. We find an expression for the Hilbert polynomial $P_Γ^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ Γ$. We also give a combinatorial description of the coefficients of $P_Γ^i(k)$.