Rooted tree graphs and the Butcher group: Combinatorics of elementary perturbation theory

TL;DR

This paper explores the combinatorial structure of perturbation series in differential equations using rooted trees, highlighting the algebraic structures like the Butcher group that organize these series.

Contribution

It clarifies the combinatorial and algebraic structures of perturbation series, emphasizing labeled rooted trees as the most straightforward framework.

Findings

Rooted trees index perturbation series terms.

The Butcher group captures composition operations on series.

Labeled rooted trees provide a simple realization framework.

Abstract

The perturbation expansion of the solution of a fixed point equation or of an ordinary differential equation may be expressed as a power series in the perturbation parameter. The terms in this series are indexed by rooted trees and depend on a parameter in the equation in a way determined by the structure of the tree. Power series of this form may be considered more generally; there are two interesting and useful group structures on these series, corresponding to operations of composition and substitution. The composition operation defines the Butcher group, an infinite dimensional group that was first introduced in the context of numerical analysis. This survey discusses various ways of realizing these rooted trees: as labeled rooted trees, or increasing labeled rooted trees, or unlabeled rooted trees. It is argued that the simplest framework is to use labeled rooted trees.