Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, that is, if an outer commutator word omega takes m different values in a group G, then the order of the verbal subgroup omega(G) is bounded by a function depending only on m, and not on omega or G. This is obtained as a consequence of a structure theorem for the subgroup omega(G), which is valid if G is soluble, and without assuming that omega takes finitely many values in G. More precisely, there is an abelian series of omega(G), such that every section of the series can be generated by values of omega all of whose powers are also values of omega in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction.