An and/or tree is a binary plane tree, with internal nodes labelled by connectives, and with leaves labelled by literals chosen in a fixed set of $k$ variables and their negations. We introduce the first model of such Catalan trees, whose number of variables $k_n$ is a function of $n$, its number of leaves. We describe the whole range of the probability distributions depending on the functions $k_n$, as soon as it tends jointly with $n$ to infinity. As a by-product we obtain a study of the satisfiability problem in the context of Catalan trees. Our study is mainly based on analytic combinatorics and extends the Kozik’s pattern theory, first developed for the fixed-$k$ Catalan tree model.