This paper gives a new, purely semantic proof of the following theorem: if an intermediate propositional logic L has the disjunction property then a disjunction free formula is provable in L iff it is provable in intuitionistic logic. The main idea of the proof is to use the well-known semantic criterion of the disjunction property for "simulating" finite binary trees (which characterize the disjunction free fragment of intuitionistic logic) by general frames.
"A New Solution to a Problem of Hosoi and Ono." Notre Dame J. Formal Logic 35 (3) 450 - 457, /Summer 1994. https://doi.org/10.1305/ndjfl/1040511350