Enumeration of algebras close to absolutely free algebras and binary trees

We study three classes of algebras: absolutely free algebras, free commutative non-associative, and free anti-commutative non-associative algebras. We study asymptotics of the growth for free algebras of these classes and for their subvarieties as well. Mainly, we study finitely generated algebras, also the codimension growth for varieties in theses classes is studied. For these purposes we use ordinary generating functions as well as exponential generating functions. The following subvarieties are studied in these classes: solvable, completely solvable, right-nilpotent, and completely right-nilpotent subvarieties. The obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees. We enumerate these trees using generating functions. For solvable and right-nilpotent algebras the generating functions are algebraic. For completely solvable and completely right-nilpotent algebras the respective functions are rational. It is known that these three varieties of algebras satisfy Schreier's property, i.e., subalgebras of free algebras are free. For free groups, there is Schreier's formula for the rank of a subgroup of a free group. We find analogues of this formula for these varieties. They are written in terms of series. As an application, we study invariants of finite groups acting on absolutely free algebras.