The number of Boolean functions computed by formulas of a given size

Estimates are given of the number B(n; L) of distinct functions computed by propositional formulas of size L in n variables, constructed using only literals and ^; _ connectives. (L is the number of occurrences of variables. L ?1 is the number of binary ^'s and _'s. B(n; L) is also the number of functions computed by two terminal series-parallel networks with L switches.) Enroute the read-once functions , which are closely related to Schrr oder numbers, are enumerated. Writing B(n; L) = b(n; L) L , we nd that if L and (n) go to innnity with increasing n and L 2 n =n (n) , then b(n; L) cn, where c = 2=(ln 4 ? 1). Making a comparison with polynomial size Boolean circuits, this implies the following. For any constant > 1, almost all Boolean functions with formula complexity at most n , cannot be computed by any circuit constructed from literals and fewer than ?1 n two input ^; _ gates.