Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees

In this paper we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in \cite{BreuerFrank2009} in the discrete case as well as for sparse trees in the metric case.