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 kn 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 (kn)n≥1. The fundamental breakthrough of this paper is to generalise the previous results for any (reasonable) sequence of integers (kn)n≥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) phenomena at kn=n/lnn.