  We demonstrate that a natural construction based on the two notions of
insertion of a strand and finding the preimages of Temperley-Lieb algebra
generators give an inductive means to generate all link patterns of a given
number of strands. It is shown that the structure of the Catalan tree (as
defined by Julian West) arises in the process of this induction, and that it
can be exploited to give some refined enumerations of link patterns.
