Subcritical pattern languages for and/or trees

Abstract

Let $P_k(f)$ denote the density of and/or trees defining a boolean function $f$ within the set of and/or trees with fixed number of variables $k$. We prove that there exists constant $B_f$ such that $P_k(f) \sim B_f \cdot k^{-L(f)-1}$ when $k \to \infty$, where $L(f)$ denote the complexity of $f$ (i.e. the size of a minimal and/or tree defining $f$). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution $\pi$ defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for $\pi$.