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 Polit├Ęcnica de Catalunya and Claas Roever of the National University of Ireland- Galway.