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We demonstrate that a natural construction based on the two notions ofinsertion of a strand and finding the preimages of Temperley-Lieb algebragenerators give an inductive means to generate all link patterns of a givennumber of strands. It is shown that the structure of the Catalan tree (asdefined by Julian West) arises in the process of this induction, and that itcan be exploited to give some refined enumerations of link patterns.

Link Patterns and the Catalan Tree