  We introduce and study a notion of decomposition of planar point sets (or
rather of their chirotopes) as trees decorated by smaller chirotopes. This
decomposition is based on the concept of mutually avoiding sets (which we
rephrase as \emph{modules}), and adapts in some sense the modular decomposition
of graphs in the world of chirotopes. The associated tree always exists and is
unique up to some appropriate constraints. We also show how to compute the
number of triangulations of a chirotope efficiently, starting from its tree and
the (weighted) numbers of triangulations of its parts.
