  In this paper, we define a new coproduct on the space of decorated planar
rooted forests to equip it with a weighted infinitesimal unitary bialgebraic
structure. We introduce the concept of $\Omega$-cocycle infinitesimal
bialgebras of weight $\lambda$ and then prove that the space of decorated
planar rooted forests $H_{\mathrm{RT}}(X,\Omega)$, together with a set of
grafting 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 an
application, we equip a free cocycle infinitesimal unitary bialgebraic
structure on the undecorated planar rooted forests, which is the object studied
in the well-known (noncommutative) Connes-Kreimer Hopf algebra. Finally, we
construct a new pre-Lie algebraic structure on decorated planar rooted forests.
