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We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions, and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients., 20 pages

In order to study some sets of probabilities, called induced averages by J. Ecalle, F. Menous introduces two grafting operators $ B^{+} $ and $ B^{-} $. With these two operators, we construct Hopf algebras of rooted and ordered trees $ \mathcal{B}^{i} $, $ i \in \mathbb{N}^{\ast} $, $ \mathcal{B}^{\infty} $ and $ \mathcal{B} $ satisfying the inclusion relations $ \mathcal{B}^{1} \subseteq \hdots \mathcal{B}^{i} \subseteq \mathcal{B}^{i+1} \subseteq \hdots \subseteq \mathcal{B}^{\infty} \subseteq \mathcal{B} $. We endow $ \mathcal{B} $ with a structure of duplicial dendriform bialgebra and we deduce that $ \mathcal{B} $ is cofree and self-dual. Finally, we introduce the notion of bigraft algebra and we prove that $ \mathcal{B} $ is generated as bigraft algebra by the element $ \tdun{1} $., Comment: In french, 32pages

We prove that the Hopf algebra of parking functions and the Hopf algebra of ordered forests are isomorphic, using a rigidity theorem for a particular type of bialgebras., Comment: 22 pages, revised version