  This paper introduces the braidings of dendriform algebras and tridendriform
algebras. By studying free braided dendriform algebras, we obtain braidings of
the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also
give a variation of the braiding of Foissy for the noncommutative
Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted
forests so that the well-known isomorphism between this Hopf algebra and the
Loday-Ronco Hopf algebra is extended to the braided context. As free braided
tridendriform algebras, we also give braided extension of the Hopf algebra of
Loday and Ronco on planar rooted trees.
