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We consider a probability distribution on the set of Boolean functions in nvariables which is induced by random Boolean expressions. Such an expression isa random rooted plane tree where the internal vertices are labelled withconnectives And and OR and the leaves are labelled with variables or negatedvariables. We study limiting distribution when the tree size tends to infinityand derive a relation between the tree size complexity and the probability of afunction. This is done by first expressing trees representing a particularfunction as expansions of minimal trees representing this function and thencomputing the probabilities by means of combinatorial counting argumentsrelying on generating functions and singularity analysis.

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees