  Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal
bialgebra is at the same time an algebra and coalgebra, in such a way that the
comultiplication is a derivation. Twenty years after Joni and Rota, Aguiar
introduced the concept of an infinitesimal (non-unitary) Hopf algebra. In this
paper we study infinitesimal unitary bialgebras and infinitesimal unitary Hopf
algebras, in contrary to Aguiar's approach. Using an infinitesimal version of
the Hochschild 1-cocycle condition, we prove respectively that a class of
decorated planar rooted forests is the free cocycle infinitesimal unitary
bialgebra and free cocycle infinitesimal unitary Hopf algebra on a set. As an
application, we obtain that the planar rooted forests is the free cocycle
infinitesimal unitary Hopf algebra on the empty set.
