  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.
