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We introduce the notions of preordered and heap-preordered forests,generalizing the construction of ordered and heap-ordered forests. We provethat the algebras of preordered and heap-preordered forests are Hopf for thecut coproduct, and we construct a Hopf morphism to the Hopf algebra of packedwords. Moreover, we define another coproduct on the preordered forests given bythe contraction of edges. Finally, we give a combinatorial description ofmorphims defined on Hopf algebras of forests with values in the Hopf algebrasof shuffes or quasi-shuffles.

Preordered forests, packed words and contraction algebras