Combinatorial Dyson-Schwinger equations, polymomial functors, and operads of Feynman graphs After briefly recalling the Connes-Kreimer Hopf algebra of Feynman graphs and the Butcher-Connes-Kreimer Hopf algebra of trees, as they appear in BPHZ renormalisation in Quantum Field Theory, a main point of this talk is to explain the relationship between the two via an intermediate bialgebra of P-trees, for P a certain polynomial functor of primitive graphs, and to outline some categorical interpretations resulting from this viewpoint. The combinatorial Dyson-Schwinger equations of Bergbauer-Kreimer take the form of polynomial fixpoint equations in groupoids, X = 1 + P(X); the solution X (which is accordingly a W-type in the sense of type theory) consists of certain nested graphs, which are the P-trees, appearing as the operations of $\bar P$, the free monad on P. The so-called overlapping divergences, a main subtlety of BPHZ renormalisation, are interpreted as the relations defining the operad of graphs as a quotient of $\bar P$. In Quantum Chromodynamics one is interested in certain truncations of the DSEs, required to generate sub-Hopf algebras. In the abstract setting, truncation is interpreted as monomorphic natural transformations between polynomial endofunctors, and the sub-Hopf condition is satisfied by the cartesian natural transformations.