AbstractLet C RF S denote the category of S-colored rooted forests, and HC RF Sdenoteits Ringel-Hall algebra as introduced in . We construct a homomorphism from aK+0(C RF S)–graded version of the Hopf algebra of noncommutative symmetric func-tions to HC RF S. Dualizing, we obtain a homomorphism from the Connes-KreimerHopf algebra to a K+0(C RF S)–graded version of the algebra of quasisymmetricfunctions. This homomorphism is a refinement of one considered by W. Zhao in ...