  We address the enumeration of p-valent planar maps equipped with a spanning
forest, with a weight z per face and a weight u per connected component of the
forest. Equivalently, we count p-valent maps equipped with a spanning tree,
with a weight z per face and a weight \mu:=u+1 per internally active edge, in
the sense of Tutte; or the (dual) p-angulations equipped with a recurrent
sandpile configuration, with a weight z per vertex and a variable \mu:=u+1 that
keeps track of the level of the configuration. This enumeration problem also
corresponds to the limit q -> 0 of the q-state Potts model on p-angulations.
Our approach is purely combinatorial. The associated generating function,
denoted F(z,u), is expressed in terms of a pair of series defined implicitly by
a system involving doubly hypergeometric series. We derive from this system
that F(z,u) is differentially algebraic in z, that is, satisfies a differential
equation in z with polynomial coefficients in z and u. This has recently been
proved to hold for the more general Potts model on 3-valent maps, but via a
much more involved and less combinatorial proof. For u >= -1, we study the
singularities of F(z,u) and the corresponding asymptotic behaviour of its n-th
coefficient. For u>0, we find the standard asymptotic behaviour of planar maps,
with a subexponential term in n^{-5/2}. At u=0 we witness a phase transition
with a term n^{-3}. When u\in[-1,0), we obtain an extremely unusual behaviour
in n^{-3}(\ln n)^{-2}. To our knowledge, this is a new "universality class" for
planar maps.
