  We study P-Hopf algebras with one coassociative cooperation over different
operads P. For example, we consider the Loday-Ronco dendriform Hopf algebra and
its isomorphisms with the noncommutative planar Connes-Kreimer Hopf algebra and
with a Hopf algebra of Brouder and Frabetti. We focus on Hopf algebra
structures over free operads, like the operad Mag freely generated by a
non-commutative non-associative binary operation, and the operad of Stasheff
polytopes.
  In order to describe the operads of primitive elements we prove an analogon
of the Poincare-Birkhoff-Witt theorem. We determine the generating series for
these operads and show that the dimension of PrimMag(n) is related to the
log-Catalan numbers. By a recursive method we show how, for small n, these
spaces can be described as modules over the symmetric groups.
