  We introduce a random probability measure on the profinite completion of the
random tree of a branching process and introduce the canonical and grand
canonical ensembles of random repelling particles on this random profinite
completion at inverse temperature $\beta > 0$. We think of this as a random
spatial process of particles in a random tree, and we introduce the notion of
the {\em mean} canonical and grand canonical partition functions where in this
context `mean' means averaged over the random environment. We give a recursion
for these mean partition functions and demonstrate that in certain instances,
determined by the law for the branching process, these partition functions as a
function of $\beta$ have algebraic properties which generalize those that
appear in the non-random and $p$-adic environments.
