Infinitesimal Bialgebra on Planar Binary Trees

We construct a weight-zero infinitesimal bialgebra structure on the $\text{bk}$-module spanned by planar binary trees, using the under product $\mathrm{\backslash }$ of Aguiar–Sottileand a root-recursive coproduct $\text{DeltaLR}$. We prove that $\text{DeltaLR}$ is coassociative and satisfies the infinitesimal derivation rule with respect to $\mathrm{\backslash }$, hence gives a unitary infinitesimal bialgebra distinct from the usual Loday–Ronco Hopf structure. We also obtain an elementary vertex-cut formula, establish freeness properties for unitary $(\mathrm{\backslash },\vee )$-algebras and unitary infinitesimal $(\mathrm{\backslash },\vee )$-bialgebras, and identify the construction with the infinitesimal coproduct transported from planar rooted forests.