  Returning to one of the original generalizations of the AKLT state, we extend
prior analysis on the Bethe lattice (or Cayley tree) to a variant with a series
of $n$ spin-1 decorations placed on each edge. The recurrence relations derived
for this system demonstrate that such systems are critical for coordination
numbers $z=3^{n+1}$, demonstrating order for greater and disorder for lesser
coordination number. We then generalize further, effectively interpolating
between systems with different values of $n$, using two realizations, one
isotropic under local $SU(2)$ transformations and one anisotropic. Exact
analysis of these recurrence relations allows us to deduce the location and
behavior of order-disorder phase transitions for $z>4$.
