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In this paper, we define a new coproduct on the space of decorated planarrooted forests to equip it with a weighted infinitesimal unitary bialgebraicstructure. We introduce the concept of $\Omega$-cocycle infinitesimalbialgebras of weight $\lambda$ and then prove that the space of decoratedplanar rooted forests $H_{\mathrm{RT}}(X,\Omega)$, together with a set ofgrafting operations $\{ B^+_\omega \mid \omega\in \Omega\}$, is the free$\Omega$-cocycle infinitesimal unitary bialgebra of weight $\lambda$ on a set$X$, involving a weighted version of a Hochschild 1-cocycle condition. As anapplication, we equip a free cocycle infinitesimal unitary bialgebraicstructure on the undecorated planar rooted forests, which is the object studiedin the well-known (noncommutative) Connes-Kreimer Hopf algebra. Finally, weconstruct a new pre-Lie algebraic structure on decorated planar rooted forests.

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles