How do algebraic datatypes relate to free structures?

I know that:

• Binary leaf trees are free magmas.
• Non-empty lists are free semigroups.
• Lists are free monoids.
• The generalized algebraic datatype data FM a where {Return::a->FM a; Bind::FM b -> (b->FM a) -> FM a} is a free monad.

I know also that various abstract datatypes also correspond to free structures—bags are free commutative monoids and finite sets are free commutative idempotent monoids.

Do other algebraic datatypes (e.g., Maybe, or 2-3 trees, or binomial trees) also correspond to free algebraic structures? If so, how might those be characterized?