Letter to letter top-down tree transducers are investigated in this paper. Informally, trees which appear in the rules of such transducers are reduced to one letter in the right-hand side as in the left one. With an encoding of the tree transformations induced by such transducers into recognizable forests, we recently established the decidability of equivalence for linear top-down transducers. Here, in order to capture the non-linearity of top-down transducers, we introduce new classes of tree automata with equivalence constraints between direct subterms for which equivalence is decidable. We then show that the equivalence problem for non-linear top-down transducers can be reduced to the equivalence problem of automata with equivalence constraints.