  Two operads are said to belong to the same Wilf class if they have the same
generating series. We discuss possible Wilf classifications of non-symmetric
operads with monomial relations. As a corollary, this would give the same
classification for the operads with a finite Groebner basis.
  Generally, there is no algorithm to decide whether two finitely presented
operads belong to the same Wilf class. Still, we show that if an operad has a
finite Groebner basis, then the monomial basis of the operad forms an
unambiguous context-free language. Moreover, we discuss the deterministic
grammar which defines the language. The generating series of the operad can be
obtained as a result of an algorithmic elimination of variables from the
algebraic system of equations defined by the Chomsky--Schutzenberger
enumeration theorem. We then focus on the case of binary operads with a single
relation. The approach is based on the results by Rowland on pattern avoidance
in binary trees. We improve and refine Rowland's calculations and empirically
confirm his conjecture. Here we use both the algebraic elimination and the
direct calculation of formal power series from algebraic systems of equations.
Finally, we discuss the connection of Wilf classes with algorithms for the
Quillen homology of operads calculation.
