  In this paper we investigate the eigenvalue statistics of exponentially
weighted ensembles of full binary trees and $p$-branching star graphs. We show
that spectral densities of corresponding adjacency matrices demonstrate
peculiar ultrametric structure inherent to sparse systems. In particular, the
tails of the distribution for binary trees share the "Lifshitz singularity"
emerging in the one-dimensional localization, while the spectral statistics of
$p$-branching star-like graphs is less universal, being strongly dependent on
$p$. The hierarchical structure of spectra of adjacency matrices is interpreted
as sets of resonance frequencies, that emerge in ensembles of fully branched
tree-like systems, known as dendrimers. However, the relaxational spectrum is
not determined by the cluster topology, but has rather the number-theoretic
origin, reflecting the peculiarities of the rare-event statistics typical for
one-dimensional systems with a quenched structural disorder. The similarity of
spectral densities of an individual dendrimer and of ensemble of linear chains
with exponential distribution in lengths, demonstrates that dendrimers could be
served as simple disorder-less toy models of one-dimensional systems with
quenched disorder.
