A hyponormal weighted shift on a directed tree whose square has trivial   domain
  It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable directed trees, one with root, the other without, whose every vertex has enumerable set of successors. 
Z. J. Jablonski, I. B. Jung and J. Stochel
math.FA