Trees and Nest Structures

This paper is a sequel to [1]. Our purpose is to exhibit certain basic relationships between (ordered dyadic) trees and regular nest structures. We introduce the notion of a tree being isomorphic to a nest structure, and we study some necessary and sufficient conditions for the existence of such isomorphisms. By virtue of some of our results, Kdnig's lemma on infinite trees, and our fundamental lemma of [1] concerning infinite regular nest structures become intimately related either yields an alternative proof of the other. The investigations of this paper arose largely out of the consideration of translation processes from proofs by semantic tableaux to proofs by natural deduction. In ? 4 we consider a variant of the semantic tableaux of Beth [2], or Hintikka [3], which we term "analytic tableaux". The translation process we need for ? 4 is best described and studied in a more general setting than the special context of quantification theory; it is basically a general combinatorial problem about trees and nest structures. Sections ? 1, ? 2, ? 3 of this paper, which are purely combinatorial in character, presuppose only ? 1 of [1].