Abstract:
Given a class of linear order types C, we identify and study several different classes of trees, naturally associated
with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and
completely determine the set-theoretic relationships between these classes of trees and between their corresponding
first-order theories. We then obtain some general results about the axiomatization of the first-order theories of
some of these classes of trees in terms of the first-order theory of the generating class C, and indicate the problems
obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable
paths in trees, that need not satisfy the first-order theory of C, so we have started analysing first-order
definable and undefinable paths in trees.
