  Building on and from the work of Brydges and Imbrie, we give an elementary
calculation of the volume of the space of branched polymers of order $n$ in the
plane and in 3-space. Our development reveals some more general identities, and
allows exact random sampling. In particular we show that a random 3-dimensional
branched polymer of order $n$ has diameter of order $\sqrt{n}$.
