  This article is motivated by the following satisfiability question: pick
uniformly at random an and/or Boolean expression of length n, built on a set of
k_n Boolean variables. What is the probability that this expression is
satisfiable? asymptotically when n tends to infinity?
  The model of random Boolean expressions developed in the present paper is the
model of Boolean Catalan trees, already extensively studied in the literature
for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this
paper is to generalise the previous results to any (reasonable) sequence of
integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above
satisfiability question.
  We also analyse the effect of introducing a natural equivalence relation on
the set of Boolean expressions. This new "quotient" model happens to exhibit a
very interesting threshold (or saturation) phenomenon at k_n = n/ln n.
