  A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie
product satisfying some compatibilities with the product and the coproduct. We
here give examples of cofree Com-PreLie bialgebras, including all the ones such
that the preLie product is homogeneous of degree $\ge$ --1. We also give a
graphical description of free unitary Com-PreLie algebras, explicit their
canonical bialgebra structure and exhibit with the help of a rigidity theorem
certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted
trees. We finally prove that the dual of these bialgebras are also enveloping
algebras of preLie algebras, combinatorially described.
