  Recent work on perturbative quantum field theory has led to much study of the
Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf
algebra of rooted trees, had already been studied by algebraists. L. Foissy
introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which
turns out to be self-dual. Using some homomorphisms defined by the author and
W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf
algebras to each other and to the Hopf algebras of symmetric functions,
noncommutative symmetric functions, and quasi-symmetric functions.
