  We introduce two operads which own the set of planar forests as a basis. With
its usual product and two other products defined by different types of
graftings, the algebra of planar rooted trees H becomes an algebra over these
operads. The compatibility with the infinitesimal coproduct of H and these
structures is studied. As an application, an inductive way of computing the
dual basis of H for its infinitesimal pairing is given. Moreover, three
Cartier-Quillen-Milnor-Moore theorems are given for the operads of planar
forests and a rigidity theorem for one of them.
