  The classical Lagrange inversion formula is extended to analytic and
non--analytic inversion problems on non--Archimedean fields. We give some
applications to the field of formal Laurent series in $n$ variables, where the
non--analytic inversion formula gives explicit formal solutions of general
semilinear 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, we
give sufficient condition for the linearization to belong to some Classes of
ultradifferentiable germs, closed under composition and derivation, including
Gevrey Classes. We prove that Bruno's condition is sufficient for the
linearization to belong to the same Class of the germ, whereas new conditions
weaker than Bruno's one are introduced if one allows the linearization to be
less regular than the germ. This generalizes to dimension $n> 1$ some results
of [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by mean
of trees, allows us to point out the strong similarities existing between the
two linearization problems, formulated (essentially) with the same functional
equation. For analytic vector fields of $\C^2$ we prove a quantitative estimate
of a previous qualitative result of [MatteiMoussu] and we compare it with a
result of [YoccozPerezMarco].
