  We consider a probability distribution on the set of Boolean functions in n
variables which is induced by random Boolean expressions. Such an expression is
a random rooted plane tree where the internal vertices are labelled with
connectives And and OR and the leaves are labelled with variables or negated
variables. We study limiting distribution when the tree size tends to infinity
and derive a relation between the tree size complexity and the probability of a
function. This is done by first expressing trees representing a particular
function as expansions of minimal trees representing this function and then
computing the probabilities by means of combinatorial counting arguments
relying on generating functions and singularity analysis.
