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Combinatorial Hopf algebras of trees exemplify the connections betweenoperads and bialgebras. Painted trees were introduced recently as examples ofhow graded Hopf operads can bequeath Hopf structures upon compositions ofcoalgebras. We put these trees in context by exhibiting them as the minimalelements of face posets of certain convex polytopes. The full face posetsthemselves often possess the structure of graded Hopf algebras (with one-sidedunit). We can enumerate faces using the fact that they are structure types ofsubstitutions of combinatorial species. Species considered here include orderedand unordered binary trees and ordered lists (labeled corollas). Some of thepolytopes that constitute our main results are well known in other contexts.First we see the classical permutohedra, and then certain generalizedpermutohedra: specifically the graph associahedra of suspensions of certainsimple graphs. As an aside we show that the stellohedra also appear as liftingsof generalized permutohedra: graph composihedra for complete graphs. Thus ourresults give examples of Hopf algebras of tubings and marked tubings of graphs.We also show an alternative associative algebra structure on the graph tubingsof star graphs.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes