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 ${\mathcal{B}}^i$, $i\in {\mathbb{N}}^{\ast }$, ${\mathcal{B}}^{\mathrm{\infty }}$ 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 $\mathcal{B}$ $\mathrm{.}Weendow$ $\mathcal{B}$ $withastructureofduplicialdendriformbialgebraandwededucethat$ $\mathcal{B}$ $ 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 $\text{tdun}1$.