Diameter minimal trees

Using the method of seeds and branch duplication, it is shown that for every tree of diameter d <7, there is an Hermitian matrix with as few as d distinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter d, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of d. Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made.