  We studied the thermodynamic behaviors of non-interacting bosons and fermions
trapped by a scale-invariant branching structure of adjustable degree of
heterogeneity. The full energy spectrum in tight-binding approximation was
analytically solved . We found that the log-periodic oscillation of the
specific heat for Fermi gas depended on the heterogeneity of hopping. Also, low
dimensional Bose-Einstein condensation occurred only for non-homogeneous setup.
