  We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$
given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an
action of the symmetric group $\Sigma_k$ by permuting the $H$-summands. We show
that the sequence of representations of the symmetric group
$H_q(\mathrm{Conf}_n(G_\bullet); \mathbf{Q})$, the homology of the ordered
configuration space of these spaces, is representation stable in the sense of
Church and Farb. In the case where $G$ and $H$ are trees, we provide a similar
result for glueing along arbitrary subtrees instead of the base point.
Furthermore, we show that stabilization alway holds for $q = 1$.
