  We provide a Cuntz-Pimsner model for graph of groups $C^*$-algebras. This
allows us to compute the $K$-theory of a range of examples and show that graph
of groups $C^*$-algebras can be realised as Exel-Pardo algebras. We also make a
preliminary investigation of whether the crossed product algebra of
Baumslag-Solitar groups acting on the boundary of certain trees satisfies
Poincar\'e duality in $KK$-theory. By constructing a $K$-theory duality class
we compute the $K$-homology of these crossed products.
