We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to - ∞ , a narrow cluster of finitely many eigenvalues tends to - ∞ , while the eigenvalues above this cluster remain bounded from below. Certain "rogue" eigenvalues break away from this cluster and tend even faster toward - ∞ . The spectrum can be visualized as the intersection points of two objects in the plane—a spiral curve depending on the Schrödinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter. [ABSTRACT FROM AUTHOR]