a set $S$ is a map $\lab\{$cycles of $\Gamma\}\rightarrow S$, we will write $c_i:=\lab^{-1}(i)$. By a labelling of a black and white tree by a set $S$ we mean a map $\lab:E_w\rightarrow S$. Again we will write $v_i:=\lab^{-1}(i)$.

### Planar planted bipartite labelled trees with white leaves

We set $\wlbptree(n)$ to be the set of planar planted bipartite trees which are labelled from $\{1,\dots,n\}$ with white leaves only. To avoid cluttered notation, we also denote the respective free Abelian group and the $k$-vector space with basis $\wlbptree(n)$ by the same name and let $\wlbptree$ be their union respectively direct sum.

Cacti
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A cactus with $n$ lobes is a $\{0,1, \dots ,n\}$ labelled marked treelike ribbon graph with a metric. I.e. The set $\Cacti(n)$ is the set of these graphs. $\Cact(n)\subset \Cacti(n)$ is the subset of spineless graphs and called the spineless cacti or alternatively cacti without spines. $\Cacti^1(n)\subset \Cacti(n)$ is the subset of normalized graphs, called normalized cacti, and finally $\Cact^1(n)=\Cact(n)\cap\Cacti^1(n)$ is the set of normalized spineless cacti.

### Cactus terminology

The edges of a cactus are traditionally called arcs or segments and the cycles of a cactus are traditionally called lobes. The vertices are sometimes called the marked 