Counting trees and matching via continuous fractions

In Combinatorics, a typical question asks to count the number of combinatorial objects of a certain kind, e.g. the number of spanning trees of perfect matchings in a given graph. In the past few years, the inverse question has also become popular, e.g. what is the smallest size graph which has a given number of spanning trees, or of a given number of perfect matchings? These questions turned out to be deeply related to classic problems and results in number theory.

In the first part of the talk I will give a brief overview of several combinatorial functions where this inverse problem has been resolved. I will then discuss a connection between two problems discussed above and Zaremba type questions and results on continued fractions. I will conclude with a discussion of our latest joint work with Chan and Kontorovich which gives best known bounds for spanning trees.