Colored trees and noncommutative symmetric functions

Let ${\text{CRF}}_S$ denote the category of $S$-colored rooted forests, and \H_{\CRF_S} denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K_0^{+}({\text{CRF}}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to \H_{\CRF_S}. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0 (\CRF_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao in \cite{Z}.