  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.
