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Quantified propositional intuitionistic logic is obtained from propositionalintuitionistic logic by adding quantifiers \forall p, \exists p overpropositions. In the context of Kripke semantics, a proposition is a subset ofthe worlds in a model structure which is upward closed. Kremer (1997) has shownthat the quantified propositional intuitionistic logic H\pi+ based on the classof all partial orders is recursively isomorphic to full second-order logic. Heraised the question of whether the logic resulting from restriction to trees isaxiomatizable. It is shown that it is, in fact, decidable. The methods used canalso be used to establish the decidability of modal S4 with propositionalquantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page

Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω