The Tree of Trees: on methods for finding all non-isomorphic tree-realizations of degree sequences

A degree sequence for a graph is a list of positive integers, one for every vertex, where each integer corresponds to the number of neighbors of that vertex. It is possible to create sequences that have no corresponding graphs, as well as sequences that correspond to multiple distinct (i.e., non-isomorphic) graphs. A graph that corresponds to a given degree sequence is called a realization of that sequence, and those degree sequences corresponding to trees are referred to as tree-realizable, or tree-able, sequences. In this thesis, we present a program that determines the number of tree realizations for any tree-able sequence, along with the sequences with the most tree-realizations (for fixed sequence lengths).