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