  Loday and Ronco defined an interesting Hopf algebra structure on the linear
span of the set of planar binary trees. They showed that the inclusion of the
Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer
Hopf algebra of permutations factors through their Hopf algebra of trees, and
these maps correspond to natural maps from the weak order on the symmetric
group to the Tamari order on planar binary trees to the boolean algebra.
  We further study the structure of this Hopf algebra of trees using a new
basis for it. We describe the product, coproduct, and antipode in terms of this
basis and use these results to elucidate its Hopf-algebraic structure. We also
obtain a transparent proof of its isomorphism with the non-commutative
Connes-Kreimer Hopf algebra of Foissy, and show that this algebra is related to
non-commutative symmetric functions as the (commutative) Connes-Kreimer Hopf
algebra is related to symmetric functions.
