Several Hopf algebra structures on vector spaces of trees can be found
in the literature (cf. Loday and Ronco, Connes and Kreimer, Brouder and
Frabetti). In this paper, we compare the corresponding notions of trees,
the multiplications and comultiplications. The Hopf algebras are connected
graded or, equivalently, complete Hopf algebras. The Hopf algebra structure
on planar binary trees introduced by Loday and Ronco is noncommutative
and not cocommutative. We show that this Hopf algebra is isomorphic to
the noncommutative version of the Hopf algebra of Connes and Kreimer. We
compute its first Lie algebra structure constants in the sense of [A Pseudo-Analyzer
Approach...], and show that there is no cogroup structure compatible with
the Hopf algebra on planar binary trees.