Lifting trees of prime ideals to Bezout extension domains

We prove that if an extension R subset of or equal to T of commutative rings satisfies the going-up property, then any tree of prime ideals of R with at most two branches or in which each branch has finite length is covered by some corresponding tree of prime ideals of T. In particular, if R subset of or equal to T is an integral extension and R is Noetherian, then each tree T in Spec(R) can be covered by a tree in Spec(T). We also prove that if R is an integral domain, then each tree T in Spec(R) can be covered by a tree in Spec(T) for some Bezout domain T containing R. If T has only finitely many branches, it can further be arranged that the Bezout domain T be an overring of R. However, in general, it cannot be arranged that T be covered rom a Prufer overring of R, thus answering negatively a question of D.D. Anderson.