A matrix algebra approach to tree shape

In this paper we investigate eigenvalues of matrix representations of trees as a means by which one might quantify the shape of a tree. We consider the adjacency matrix, the Laplacian matrix, and the pairwise distance matrix of the tree. We then demonstrate that for any of these choices of matrix the fraction of trees with a unique set of eigenvalues goes to zero as the number of leaves goes to infinity. This implies that the eigenvalues of these matrix representations may not give as much information as might be hoped. We investigate the rate of convergence of the above fraction to zero using numerical methods.