  We study grand--canonical and canonical properties of the model of branched
polymers proposed in \cite{adfo}. We show that the model has a fourth order
phase transition and calculate critical exponents. At the transition the
exponent $\gamma$ of the grand-canonical ensemble, analogous to the string
susceptibility exponent of surface models, $\gamma \sim 0.3237525...$ is the
first known example of positive $\gamma$ which is not of the form $1/n,\,
n=2,3,\ldots$. We show that a slight modification of the model produces a
continuos spectrum of $\gamma$'s in the range $(0,1/2]$ and changes the order
of the transition.
