Rooted Tree Graphs and Transformation Groups

Series solutions of ordinary differential equations are indexed by rooted tree graphs. It was discovered by Butcher that there is an infinite-dimensional transformation group associated with rooted tree graphs. His discovery was in the context of numerical analysis of ordinary differential equations, but the same group emerges in the Connes-Kreimer approach to renormalization in quantum field theory. This talk will explore the combinatorial background to this subject.