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.

翻译：我们引入并研究一种将平面点集（更确切地说，是其手性拓扑）分解为以小子手性拓扑装饰的树结构的概念。该分解基于互避集概念（我们将其重新表述为“模块”），并在某种意义上将图论中的模块分解思想适配到手性拓扑领域。此类关联树在适当约束下总是存在且唯一。我们还展示了如何从该树及其各组成部分的（加权）三角剖分数出发，高效计算任意手性拓扑的三角剖分数。