  As a follow-up of previous work of the authors, we analyse the statistical
mechanics model of random spanning forests on random planar graphs. Special
emphasis is given to the analysis of the critical behaviour. Exploiting an
exact relation with a model of O(-2)-loops and dimers, previously solved by
Kostov and Staudacher, we identify critical and multicritical loci, and find
them consistent with recent results of Bousquet-M\'elou and Courtiel. This is
also consistent with the KPZ relation, and the Berker-Kadanoff phase in the
anti-ferromagnetic regime of the Potts Model on periodic lattices, predicted by
Saleur. To our knowledge, this is the first known example of KPZ appearing
explicitly to work within a Berker-Kadanoff phase. We set up equations for the
generating function, at the value t=-1 of the fugacity, which is of
combinatorial interest, and we investigate the resulting numerical series, a
favourite problem of Tony Guttmann's.
