In the study of eigenvalues, multiplicities, and graphs, the minimum number of multiplicities equal to 1 in a real symmetric matrix with graph G , U ( G ), is an important constraint on the possible multiplicity lists among matrices in 𝒮( G ). Of course, the structure of G must determine U ( G ), but, even for trees, this linkage has proven elusive. If T is a tree, U ( T ) is at least 2, but may be much greater. For linear trees, recent work has improved our understanding. Here, we consider nonlinear trees, segregated by diameter. This leads to a new combinatorial construct called a core, for which we are able to calculate U ( T ). We suspect this bounds U ( T ) for all nonlinear trees with the given core. In the process, we develop considerable combinatorial information about cores.