Tree transducers (automata which read finite labeled trees and output finite labeled trees) are used to define a hierarchy of families of “tree languages” (sets of trees). In this hierarchy, families generated by “top-down” tree transducers (which read trees from the root toward the leaves) alternate with families generated by “bottom-up” tree transducers (which read trees from the leaves toward the root). A hierarchy of families of string languages is obtained from the first hierarchy by the “yield” operation (concatenating the labels of the leaves of the trees). Both hierarchies are conjectured to be infinite, and some results are presented concerning this conjecture. A study is made of the closure properties of the top-down and bottom-up families in the hierarchies under various tree and string operations. The families are shown to be closed under certain operations if and only if the hierarchies are finite.