Richard Thompson's group F is a widely studied group which has provided examples of and counter-examples to a variety of conjectures in group theory. It is also an extremely combinatorially appealing object which has a beautiful description in terms of binary trees. In this talk I will describe two important enumerative problems associated with F. The first is the problem of computing the growth-series of F - the number of elements with geodesic length n. I will describe the polynomial time algorithm that "solves" the problem and a couple of associated conjectures. The second problem is the cogrowth series - the number of words of length n equivalent to the identity. This second problem important because of its connections to the amenability of F and I will describe some of our recent experimental explorations of this problem using techniques from enumeration and statistical mechanics.