On the duality between trees and disks

A combinatorial category Disk was introduced by André Joyal to play a role in his definition of weak ω-category. He defined the category Θ to be dual to Disks. In the ensuing literature, a more concrete description of Θ was provided. In this paper we provide another proof of the dual equivalence and introduce various categories equivalent to Disk or Θ, each providing a helpful viewpoint.