Alg\`ebres de greffes

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 $\tdun{1}$.