In this paper we utilize a certain "double tree construction" to show that every "expansion", namely "universal finite type invariant (UFTI)" of parenthesized braids extends uniquely first to an expansion/UFTI of knotted trivalent graphs (a well known result), and then on to an expansion/UFTI of w-knotted objects, namely to knottings of "2-dimensional foams" and various associated objects in four-dimensioanl space.

In algebraic language, an expansion for parenthesized braids is the same as a "Drinfel'd associator" Φ, and an expansion for w-knotted objects is the same as a solution V of the Kashiwara-Vergne problem [KV] as reformulated by Alekseev and Torossian [AT]. Hence our result amounts to a topological re-interpretation of the result of Alkeseev-Enriquez-Torossian [AET] that "there is a formula for V in terms of Φ", along with an independent topological proof that the said formula indeed works - that the equations satisfied by V follow from the equations satisfied by Φ.