  An and/or tree is usually a binary plane tree, with internal nodes labelled
by logical connectives, and with leaves labelled by literals chosen in a fixed
set of k variables and their negations. In the present paper, we introduce the
first model of such Catalan trees, whose number of variables k_n is a function
of n, the size of the expressions. We describe the whole range of the
probability distributions depending on the function 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.
