  The Tamari lattice and the associahedron provide methods of measuring
associativity on a line. The real moduli space of marked curves captures the
space of such associativity. We consider a natural generalization by
considering the moduli space of marked particles on the Poincar\'{e} disk,
extending Tamari's notion of associativity based on nesting. A geometric and
combinatorial construction of this space is provided, which appears in
Kontsevich's deformation quantization, Voronov's swiss-cheese operad, and
Kajiura and Stasheff's open-closed string theory.
