Sequences of Trees and Higher-Order Renormalization Group Equations

We define a notion of higher order renormalization group equation and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a $k$th order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a $k$th order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees which require special choices of Feynman rules in order to satisfy a higher order renormalization group equation.