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We study basic spectral features of graph Laplacians associated with a classof rooted trees which contains all regular trees. Trees in this class can begenerated by substitution processes. Their spectra are shown to be purelyabsolutely continuous and to consist of finitely many bands. The main resultgives stability of absolutely continuous spectrum under sufficiently smallradially label symmetric perturbations for non regular trees in this class. Insharp contrast, the absolutely continuous spectrum can be completely destroyedby arbitrary small radially label symmetric perturbations for regular trees inthis class.

On the spectral theory of trees with finite cone type