A binary operation on trees and an initial algebra characterization for finite tree types

A binary operation on the class of trees is defined that generates a set B of finite trees form a trivial tree (one node) and B contains for every finite tree G exactly one element isomorphic to G. The binary operation defines an algebraic structure on B, and as a consequence the finite tree types are characterized as an initial algebra in the same way as the natural numbers are characterized as an initial algebra by the Peano-Lawvere axiom [2]. Simple and primitive recursion are defined and some applications of the initial algebra characterization are given.