declaring $\del(\mk(c_0))$ to be the root vertex and declaring $\mk(c_0)$ to be the smallest element makes it into a planted tree.

An equivalent definition is given by defining that there is an edge between a pair of a black and a white vertex if and only if the vertex corresponding to $b$ is on the boundary of the cycle $c_i$, i.e. $v\in \del(c_i):= \{\del(f):f\in c_i\}$.

### Spineless marked ribbon graphs {#spinlessgraph}

A marked treelike ribbon graph is called [*spineless*]{}, if

-   There is at most one vertex of valence $2$. If there is such a vertex $v_0$ then $\del(mk(c_0))=v_{0}$.

-   The induced linear orders on the $c_i$ are compatible with that of $c_0$, i.e. $f\prec_i f'$ if and only if $\imath(f')\prec_0 \imath(f)$.

### Graphs with a metric

A metric $w_{\Gamma}$ for a graph is a map $E_{\Gamma}\rightarrow
\mathbb{R}_{>0}$. The (global) re-scaling of a metric $w$ by $\lambda$ is the metric $ \lambda w: \lambda w(e)=\lambda w(e)$. The length of a cycle $c$ is the sum of the lengths of its edges $length(c)=\sum_{f\in c} w(\{f,\imath(f)\})$. A metric for a treelike ribbon graph is called normalized if the length of each non-distinguished cycle is $1$.

### Marked ribbon graphs with metric and 