Generating functions for ternary algebras and ternary trees

In this paper we consider the ternary algebras, i.e., algebras with a trilinear operation. In this class we study finitely generated algebras and their growth, as well as the growth of codimensions of absolutely free algebras and some other varieties. To this end we use ordinary generating functions and exponential generating functions (the complexity functions).
In classes of absolutely free, free symmetric, free anti-symmetric, and some other algebras we study the left-nilpotent and completely left-nilpotent algebras and subvarieties. Our results are equivalent to the enumeration of ternary trees that do not contain some forbidden subtrees of special sort.
As the main result, we prove that for varieties of left-nilpotent and completely left-nilpotent ternary algebras the complexity functions are algebraic.