Let A be a real symmetric matrix whose graph is a tree, T. If T is a linear tree (meaning all vertices with degree 3 or larger lie on the same induced path), then we can use a ”Linear Superposition Principle” to determine all possible multiplicities of eigenvalues of A. If T is a nonlinear tree, we must use other ad hoc methods. I utilize these methods to compute all possible multiplicity lists of trees on 12 vertices, and augment an existing multiplicities database. This database allows us to examine of the effects that the structure of tree can have on a multiplicity list. Then, I investigate the enumeration of linear and nonlinear trees, and examine the ratio of nonlinear trees to total trees
