An answer to a Dobbs conjecture about treed domains

We prove that a locally Jaffard integrally closed domain is such that each overring is treed if and only if it is a Prüfer domain. It follows that an integrally closed domain with valuative dimension two such that each overring is treed is necessarily going-down. This solves the long-standing open question raised by D.E. Dobbs in [D.E. Dobbs, On treed overrings and going down domains, Rend. Math. 7 (1987) 317–322]. Further applications are given.