  Since the 90's, several authors have studied a probability distribution on
the set of Boolean functions on $n$ variables induced by some probability
distributions on formulas built upon the connectors $And$ and $Or$ and the
literals $\{x_{1}, \bar{x}_{1}, \dots, x_{n}, \bar{x}_{n}\}$. These formulas
rely on plane binary labelled trees, known as Catalan trees. We extend all the
results, in particular the relation between the probability and the complexity
of a Boolean function, to other models of formulas: non-binary or non-plane
labelled trees (i.e. Polya trees). This includes the natural tree class where
associativity and commutativity of the connectors $And$ and $Or$ are realised.
