Unbounded subnormal weighted shifts on directed trees. II
  Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to subnormality of weighted shifts on directed trees is revealed. 
Piotr Budzy\'nski, Zenon Jan Jab{\l}o\'nski, Il Bong Jung, Jan Stochel
math.FA