level one can translate the respective relations from the relations in the $\Arc$ operad using the method described in [@cact; @KLP]. As it turns out, in order to translate the relations and thus to establish the homotopy BV structure on the chain level, one needs a refinement of the cell decomposition on the semi-direct product to be able to accommodate all the operations which were used in the $\Arc$ operad picture. This refinement uses cell decompositions on the $S^1$ factors which are induced by regarding them as the lobe they represent. This leads to a combinatorial description in terms of planar planted black and white (b/w) bipartite trees with additional data called spines. In the language of cacti [@cact], the additional data keeps track of the position of the local zeros. On these trees, there are linear orders at each vertex, which may differ from the induced linear order of the planar planted trees. This forces us to look at non–rooted trees or equivalently to invert the orientation of edges. According to the general calculus for “correlation functions” defined by trees, to achieve such an inversion one needs to have a non–degenerate pairing, which is symmetric and invariant. This is 