This article is concerned with semantic methods for higher-order verification. Given an alternating parity automaton A, we introduce a variant of the Scott semantics of linear logic extended with a colouring modality and a fixed point operator. We prove that the interpretation of a higher-order recursion scheme in the resulting model of the lambdaY-calculus is the set of states from which the tree it generates is accepted by A. The finiteness of the model therefore leads to a new proof of the decidability of MSO formulas at the root of trees generated by higher-order recursion schemes. By using a connection between the denotations in this model and an appropriate type system, together with the finiteness of the semantics, we also obtain a new decidability proof of the selection problem.