  The matrix model of random surfaces with c = inf. has recently been solved
and found to be identical to a random surface coupled to a q-states Potts model
with q = inf. The mean field-like solution exhibits a novel type of tree
structure. The natural question is, down to which--if any--finite values of c
and q does this behavior persist? In this work we develop, for the Potts model,
an expansion in the fluctuations about the q = inf. mean field solution. In the
lowest--cubic--non-trivial order in this expansion the corrections to mean
field theory can be given a nice interpretation in terms of structures (trees
and ``galaxies'') of spin clusters. When q drops below a finite q_c, the
galaxies overwhelm the trees at all temperatures, thus suppressing mean field
behavior. Thereafter the phase diagram resembles that of the Ising model, q=2.
