Ordered multiplicity lists for eigenvalues of symmetric matrices whose graph is a linear tree

We consider the class of trees for which all vertices of degree at least 3 lie on a single induced path of the tree. For such trees, a new superposition principle is proposed to generate all possible ordered multiplicity lists for the eigenvalues of symmetric (Hermitian) matrices whose graph is such a tree. It is shown that no multiplicity lists other than these can occur and that for two subclasses all such lists do occur. Important contrasts with trees outside the class are given, and it is shown that several prior conjectures about multiplicity lists, including the Degree Conjecture, follow from our superposition principle.