  A standard method for designing randomized algorithms to approximately count
the number of solutions of a problem in $\#$P, is by constructing a rapidly
mixing Markov chain converging to the uniform distribution over this set of
solutions. This construction is not always an easy task, and it is conjectured
that it is not always possible. We want to investigate other possibilities for
using Markov Chains in relation to counting, and whether we can relate
algorithmic counting to other, non-uniform, probability distributions over the
set we want to count.
  In this paper we present a family of probability distributions over the set
of solutions of a problem in TotP, and show how they relate to counting;
counting is equivalent to computing their normalizing factors. We analyse the
complexity of sampling, of computing the normalizing factor, and of computing
the size support of these distributions. The latter is also equivalent to
counting. We also show how the above tasks relate to each other, and to other
problems in complexity theory as well.
  In short, we prove that sampling and approximating the normalizing factor is
easy. We do this by constructing a family of rapidly mixing Markov chains for
which these distributions are stationary. At the same time we show that exactly
computing the normalizing factor is TotP-hard. However the reduction proving
the latter is not approximation preserving, which conforms with the fact that
TotP-hard problems are inapproximable if NP $\neq$ RP.
  The problem we consider is the Size-of-Subtree, a TotP-complete problem under
parsimonious reductions. Therefore the results presented here extend to any
problem in TotP. TotP is the Karp-closure of self-reducible problems in $\#$P,
having decision version in P.
