cells are indexed by trees with $k-i$ white edges and $i$ vertices marked by $1$.

Moreover cellular chains are a chain model for the framed little discs operad and form an operad. This operad is isomorphic to the semi–direct product of the chain model of the little discs operad given by $CC_*(\cact)$ of [@del] and the cellular chains of the operad built on the monoid $S^1$.

For the CW decomposition we note that as spaces $\cacti^1(n)=
\cact^1(n) \times (S^1)^{\times n}$ see Remark \[setrem\]. Now viewing $S^1=[0,1]/0\sim1$ as a 1-cell together with the 0-cell given by $0\in S^1$ the first part of the proposition follows immediately, by viewing the decoration by 1 as indicating the presence of the 1-cell of $S^1$ for that labelled component in the product of cells.

To show that the cellular chains indeed form an operad, we use the fact that the bi–crossed product is homotopy equivalent to the semi–direct product in such a way, that the action of a cell $S^1$ in the bi–crossed product is homotopic to the diagonal action. This is just the observation that the diagonal and the diagonal defined by a cactus are homotopic. Since a semi-direct product of a monoid with 