  In this paper, we first prove that a Rota-Baxter family algebra indexed by a
semigroup induces an ordinary Rota-Baxter algebra structure on the tensor
product with the semigroup algebra. We show that the same phenomenon arises for
dendriform and tridendriform family algebras. Then we construct free dendriform
family algebras in terms of typed decorated planar binary trees. Finally, we
generalize typed decorated rooted trees to typed valently decorated Schr\"oder
trees and use them to construct free tridendriform family algebras.
