  Many-body states whose wave-function admits a representation in terms of a
uniform binary-tree tensor decomposition are shown to obey to power-law
two-body correlations functions. Any such state can be associated with the
ground state of a translational invariant Hamiltonian which, depending on the
dimension of the systems sites, involve at most couplings between
third-neighboring sites. A detailed analysis of their spectra shows that they
admit an exponentially large ground space.
