Thompson's group F is a perplexing infinite finitely-presented group with a number of different ways of understanding it. F can be understood algebraically, via generators and relations with a useful normal form. F can be understood combinatorially, in terms of pairs of rooted binary trees. And F can be understood analytically, as a group of piecewise-linear homeomorphisms of an interval or as a group of maps between Cantor sets. Usually tree pair diagrams used in conjunction with F are finite, but there are some applications of infinite but periodic tree pair diagrams to understanding finite index subgroups of F and finite extensions of F. These lead to effective understanding of the automorphism group of F in a way that leads to a good description of the abstract commensurator group of F. This is joint work with Jose Burillo of the Universitat Politècnica de Catalunya and Claas Roever of the National University of Ireland- Galway.