  Butcher series appear when Runge-Kutta methods for ordinary differential
equations are expanded in power series of the step size parameter. Each term in
a Butcher series consists of a weighted elementary differential, and the set of
all such differentials is isomorphic to the set of rooted trees, as noted by
Cayley in the mid 19th century. A century later Butcher discovered that rooted
trees can also be used to obtain the order conditions of Runge-Kutta methods,
and he found a natural group structure, today known as the Butcher group. It is
now known that many numerical methods also can be expanded in Butcher series;
these are called B-series methods. A long-standing problem has been to
characterize, in terms of qualitative features, all B-series methods. Here we
tell the story of Butcher series, stretching from the early work of Cayley, to
modern developments and connections to abstract algebra, and finally to the
resolution of the characterization problem. This resolution introduces
geometric tools and perspectives to an area traditionally explored using
analysis and combinatorics.
