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

The infinitesimal Hopf algebra and the operads of planar forests