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Butcher series, also called B-series, are a type of expansion, fundamental inthe analysis of numerical integration. Numerical methods that can be expandedin B-series are defined in all dimensions, so they correspond to\emph{sequences of maps}---one map for each dimension. A long-standing problemhas been to characterise those sequences of maps that arise from B-series. Thisproblem is solved here: we prove that a sequence of smooth maps between vectorfields on affine spaces has a B-series expansion if and only if it is\emph{affine equivariant}, meaning it respects all affine maps between affinespaces.

B-series methods are exactly the affine equivariant methods