Abstract
The conditions for a B-series to be a Hamiltonian vector field imply that it may be given as a series indexed by free trees, i.e. trees without root. At present, the pre-Lie structure of rooted trees plays an important role in the study of numerical methods for ordinary differential equations, as does the associated Lie bracket on rooted trees obtained by antisymmetrization. We give a substitute for this Lie bracket defined on free trees that reflects the Lie bracket of Hamiltonian B-series, and illustrate an application of this to the backward error analysis of symplectic numerical integrators.
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