Phylogenetic trees have been used for decades to visualize evolutionary relationships graphically. Comparing topologies of trees is essential to many research areas of biology, but is complicated due to their combinatorial nature and the number of possible topologies that increases with the number of species. We will focus on binary rooted phylogenetic trees with unit edge length. Comparison can be facilitated by investigating the spectra - the set of eigenvalues - of the pairwise distance matrices of these trees. A pair of distinct trees on 17 leaves with equal spectrum already showed that spectra are not unique for the topology of the tree, however, they reveal some (sub-)structures.

In this thesis, we show that eigenvalue -2 with corresponding eigenvector v =-e_i+e_j reveals the presence of a cherry (two current species sharing a parent). Moreover, we prove that perfectly balanced trees have negative spectrum of the closed-form -2(2^k-1), where k is a number between 1 and the height of the tree. In addition, we show that an eigenvalue lambda of a submatrix appears in the spectrum of the full matrix, as long as the part of the matrix linking the subtree to the rest of the tree is orthogonal to the submatrix’s eigenvector corresponding to lambda. The remaining eigenvalues of the full matrix can be computed using Schur's formula. Finally, we combine all these results and explain the spectral equivalence in the pair of distinct trees on 17 leaves. We observe that other eigenvalues that appear in this spectrum might reveal another type of subtree.