Phase transition in fluctuating branched geometry

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