factors of $\mathbb{R}_{>0}$ are given by associating to each lobe its length. In both cases the inverse map is clear.

The topological type of a spineless cactus in $\cact^1(n)$ is defined to be its dual b/w tree $\t \in \wlbptree(n)$.

\[arctoedge\] Notice that the arcs of a cactus correspond to the set $E_{arcs}=E(\t)\setminus (\{e_{root}\})$. This bijection can be defined as follows. To a given $e\in E_{arcs}, e=\{w,b\}$ with $b$ black and $w$ white, we associate the unique arc between the points corresponding to the black vertices $b$ and $b-$ where $b-$ is the black vertex immediately preceding $b$ in the cyclic order of $v$. In other words if $e=\{f,\imath(f)\}$ with $f\in F_v$. Let $f-$ be the flag immediately preceding $f$ in the cyclic order at $v$, then $b-=\del(\imath(f-))$. Notice that if $|v|=0$ then and only then $f-=f$.

\[typelemma\] A spineless cactus is uniquely determined by its topological type and the lengths of the segments.

The CW complex of normalized spineless cacti
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We recall from [@del] the CW complexes $\CWcact(n)$. For more details and pictures the reader is referred to [@del; @cact].

\[lengthrem\] For a normalized spineless cactus the lengths of the arcs have to sum up to the radius 