Laver trees in the generalized Baire space

We prove that any suitable generalization of Laver forcing to the space κ^κ, for uncountable regular κ, necessarily adds a Cohen κ-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if κ^<κ = κ, then every <κ-distributive tree forcing on κ^κ adding a dominating κ-real which is the image of the generic under a continuous function in the ground model, adds a Cohen κ-real. This is a contribution to the study of generalized Baire spaces and answers a question from [1].