  In this paper, we first construct the free Rota-Baxter family algebra
generated by some set $X$ in terms of typed angularly $X$-decorated planar
rooted trees. As an application, we obtain a new construction of the free
Rota-Baxter algebra only in terms of angularly decorated planar rooted trees
(not forests), which is quite different from the known construction via
angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We
then embed the free dendriform (resp. tridendriform) family algebra into the
free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove
that the free Rota-Baxter family algebra is the universal enveloping algebra of
the free (tri)dendriform family algebra.
