Commutator equations in free groups

Let f ₁, ..., f _n be free generators of a free group F. We consider the equation [z ₁, ..., z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, ..., z _n]_ω and [f ₁, ..., f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solution z ₁=r₁, ..., z _z=r _n we have <r ₁, ..., r _n>=〈f ₁, ...f _n〉.