  We construct three new combinatorial Hopf algebras based on the Loday-Ronco
operations on planar binary trees. The first and second algebras are defined on
planar trees and labeled planar trees extending the Loday-Ronco and
Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is
bidendriform which implies that is also free, cofree, and self-dual. The third
algebra involves a new visualization of parking functions as decorated binary
trees; it is also bidendriform, free, cofree, and self-dual, and therefore
abstractly isomorphic to the algebra PQSym of Novelli and Thibon.
  We define partial orders on the objects indexing each of these three Hopf
algebras, one of which, when restricting to (m+1)-ary trees, coarsens the
m-Tamari order of Bergeron and Pr\'eville-Ratelle. We show that multiplication
of dual fundamental basis elements are given by intervals in each of these
orders.
  Finally, we use an axiomatized version of the techniques of Aguiar and
Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on
each of our Hopf algebras, and to show that comultiplication is cofree on the
monomial elements. This in particular, implies the cofreeness of the Hopf
algebra on planar trees. We also find explicit positive formulas for the
multiplication on monomial basis and a cancellation-free and grouping-free
formula for the antipode of monomial elements.
