  A simple and elementary proof of Butcher's theorem on the order conditions of
Runge-Kutta methods is presented. It is based on a recursive definition of
rooted trees and avoids combinatorial tools such as labelings and Faa di
Bruno's formula. This strictly recursive approach can easily and elegantly be
implemented using modern computer algrebra systems like Mathematica. The full
but short source code is included and applied to some instructive examples.
