  In this paper we introduce a kind of "noncommutative neighbourhood" of a
semiclassical parameter corresponding to the Planck constant. This construction
is defined as a certain filtered and graded algebra with an infinite number of
generators indexed by planar binary leaf-labelled trees. The associated graded
algebra (the classical shadow) is interpreted as a "distortion" of the algebra
of classical observables of a physical system. It is proven that there exists a
q-analogue of the Weyl quantization, where q is a matrix of formal variables,
which induces a nontrivial noncommutative analogue of a Poisson bracket on the
classical shadow.
