On Deening Functions on Binary Trees 1

This article is a continuation of an article on deening functions on trees (see 7]). In this article we develop terminology specialized for binary trees, rst deening binary trees and binary grammars. We recast the induction principle for the set of parse trees of binary grammars and the scheme of deening functions inductively with the set as domain. We conclude with deening the scheme of deening such functions by lambda-like expressions. 7] provide the notation and terminology for this paper. Let D be a non empty set and let t be a tree decorated with elements of D. The root label of t is an element of D and is deened by: (Def. 1) The root label of t = t("): We now state two propositions: (1) Let D be a non empty set and t be a tree decorated with elements of D. Then the roots of hti = hthe root label of ti: (2) Let D be a non empty set and t 1 , t 2 be trees decorated with elements of D. Then the roots of ht 1 ; t 2 i = hthe root label of t 1 , the root label of t 2 i: Let I 1 be a tree. We say that I 1 is binary if and only if: (Def. 2) For every element t of I 1 such that t = 2 Leaves(I 1) holds succ t = ft h0i; t h1ig: We now state several propositions: (3) For every tree T and for every element t of T holds t 2 Leaves(T) ii t h0i = 2 T: (4) For every tree T and for every element t of T holds t 2 Leaves(T) ii it is not true that there exists a natural number n such that t hni 2 T: (5) For every tree T and for every element t of T holds succ t = ; ii t 2 Leaves(T): (6) The elementary tree of 0 is binary.