  Forest formulas that generalize Zimmermann's forest formula in quantum field
theory have been obtained for the computation of the antipode in the dual of
enveloping algebras of pre-Lie algebras. In this work, largely motivated by
Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same
ideas and techniques generalize and provide effective tools to handle
computations in these algebras, which are of utmost importance in numerical
analysis and related areas. We illustrate our results by studying the action of
the pre-Lie exponential and the Magnus operator in the free pre-Lie algebra and
in a pre-Lie algebra of words originating in free probability. The latter
example provides combinatorial formulas relating the different brands of
cumulants in non-commutative probability.
