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The classical Lagrange inversion formula is extended to analytic andnon--analytic inversion problems on non--Archimedean fields. We give someapplications to the field of formal Laurent series in $n$ variables, where thenon--analytic inversion formula gives explicit formal solutions of generalsemilinear differential and $q$--difference equations.  We will be interested in linearization problems for germs of diffeomorphisms(Siegel center problem) and vector fields. In addition to analytic results, wegive sufficient condition for the linearization to belong to some Classes ofultradifferentiable germs, closed under composition and derivation, includingGevrey Classes. We prove that Bruno's condition is sufficient for thelinearization to belong to the same Class of the germ, whereas new conditionsweaker than Bruno's one are introduced if one allows the linearization to beless regular than the germ. This generalizes to dimension $n> 1$ some resultsof [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by meanof trees, allows us to point out the strong similarities existing between thetwo linearization problems, formulated (essentially) with the same functionalequation. For analytic vector fields of $\C^2$ we prove a quantitative estimateof a previous qualitative result of [MatteiMoussu] and we compare it with aresult of [YoccozPerezMarco].

The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations