Link Patterns and the Catalan Tree

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.