  The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy
Hopf algebra of of planar rooted trees are related to each other and to the
well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair
of commutative diagrams. We show how this point of view can simplify
computations in the Connes-Kreimer Hopf algebra and its dual, particularly for
combinatorial Dyson-Schwinger equations.
