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Nested parentheses are forms in an algebra which define orders ofevaluations. A class of well-formed sets of associated opening and closingparentheses is well studied in conjunction with Dyck paths and Catalan numbers.Nested parentheses also represent cuts through circles on a line. These becometopologies of non-intersecting circles in the plane if the underlying algebrais commutative.  This paper generalizes the concept and answers quantitatively - asrecurrences and generating functions of matching rooted forests - thequestions: how many different topologies of nested circles exist in the planeif (i) pairs of circles may intersect, or (ii) even triples of circles mayintersect. That analysis is driven by examining the symmetry properties of theinner regions of the fundamental type(s) of the intersecting pairs and triples.

Topologically Distinct Sets of Non-intersecting Circles in the Plane