  We first prove that a graded, connected, free and cofree Hopf algebra is
always self-dual; then that two graded, connected, free and cofree Hopf
algebras are isomorphic if, and only if, they have the same Poincar\'e-Hilbert
formal series. If the characteristic of the base field is zero, we prove that
the Lie algebra of the primitive elements of such an object is free, and we
deduce a characterization of the formal series of free and cofree Hopf algebras
by a condition of growth of the coefficients. We finally show that two graded,
connected, free and cofree Hopf algebras are isomorphic as (non graded) Hopf
algebras if, and only if, the Lie algebra of their primitive elements have the
same number of generators.
