Regions surrounded by circles whose Poincar\'e-Reeb graphs are trees

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb graphs are trees and investigate the trees obtained by a certain inductive rule from a disk in the plane. The Poincar\'e-Reeb graph of such a region is a graph whose underlying set is the set of all components of level sets of the restriction of the canonical projection to the closure and whose vertices are points corresponding to the components containing {\it singular} points. Related studies were started by the author, motivated by importance and difficulty of explicit construction of a real algebraic map onto a prescribed closed region in the plane.