  We give a natural monomorphism from the necklace Lie coalgebra, defined for
any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to
a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's
renormalization Hopf algebra, as well as to pre-Lie versions. These results are
direct analogues of Turaev's results in 2004, by replacing algebras of loops on
surfaces with algebras of paths on quivers. We also factor the morphism through
an algebra of chord diagrams and explain the geometric version. We then explain
how all of the Hopf algebras are uniquely determined by the pre-Lie structures,
and discuss noncommutative versions of the Hopf algebras.
