  This paper will describe how combinatorial interpretations can help us
understand the algebraic structure of two aspects of perturbative quantum field
theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman
graphs. The particular examples which will be looked at are, a better reduction
to geometric series for Dyson-Schwinger equations, a subgraph which yields
extra denominator reductions in scalar Feynman integrals, and an explanation of
a trick of Brown and Schnetz to get one extra step in the denominator reduction
of an important particular graph.
