The identity map on ySym makes PSym into a connection on ySym.
As PSym is graded and connected, it has a one-sided antipode.
Theorem 5.1 There are unit and antipode maps [mu] : K [right arrow] PSym and S : PSym [right arrow] PSym making PSym a one-sided Hopfalgebra.
The ySym-Hopf module structure on PSym from Theorem 4.1 has coaction
Since painted trees and bi-leveled trees both index vertices of the multiplihedra, these structures for PSym give structures on the linear span [MSym.sub.+] of bi-leveled trees with at least one node.