  We present an analytical method, based on a real space decimation scheme, to
extract the exact eigenvalues of a macroscopically large set of pinned
localized excitations in a Cayley tree fractal network. Within a tight binding
scheme we exploit the above method to scrutinize the effect of a deterministic
deformation of the network, first through a hierarchical distribution in the
values of the nearest neighbor hopping integrals, and then through a radial
Aubry Andre Harper quasiperiodic modulation. With increasing generation index,
the inflating loop less tree structure hosts pinned eigenstates on the
peripheral sites that spread from the outermost rings into the bulk of the
sample, resembling the spread of a forest fire, lighting up a predictable set
of sites and leaving the rest unignited. The penetration depth of the envelope
of amplitudes can be precisely engineered. The quasiperiodic modulation yields
hitherto unreported quantum butterflies, which have further been investigated
by calculating the inverse participation ratio for the eigenstates, and a
multifractal analysis. The applicability of the scheme to photonic fractal
waveguide networks is discussed at the end.
