On homomorphisms between products of median algebras

Homomorphisms of products of median algebras are studied with particular attention to the case when the codomain is a tree. In particular, we show that all mappings from a product ({mathbf{A_1} timesldotstimes {mathbf{A}_{n}}}) of median algebras to a median algebra ({mathbf{B}}) are essentially unary whenever the codomain ({mathbf{B}}) is a tree. In view of this result, we also characterize trees as median algebras and semilattices by relaxing the defining conditions of conservative median algebras.