We study trees T of height at most omega-1 with no uncountable
branches, and their applications in the study of pairs (A,B)
of non-isomorphic structures over a fixed vocabulary.
There is a natural quasi-ordering of such trees in terms of
the existence of a strictly increasing mapping from one tree
to another. We investigate in depth the structure of this
quasi-ordering and relate its properties to properties of pairs
(A,B) of structures. Many new constructions of pairs
of highly equivalent non-isomorphic structures are given.
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