call the set $V_w:=\color^{-1}(1)$ the white vertices. If $(f,\imath (f))$ is a naturally oriented edge, we call the edge white if $\del(\imath(f))\in V_w$ and denote the set of white edges by $E_w$. Likewise we call $V_b:=\color^{-1}(0)$ the black vertices and let $E_b$ be the set of black edges, where a naturally oriented edge $(f,\imath (f))$ is called black if $\del(\imath(f))\in V_b$.

The black leaves in a rooted black and white tree are called tails. The edges incident to the tails are called tail edges and are denoted $E_{tail}$. For tails, we will only consider those flags of the tail edges which are not incident to the tail vertices and call them $F_{tail}$.

### Planar trees and Ribbon graphs

A ribbon graph is a connected graph whose vertices are of valence at least two together with a cyclic order of the set of flags of the vertex $v$ for every vertex $v$.

A graph with a cyclic order of the flags at each vertex gives rise to bijections $N_v:F_v\rightarrow F_v$ where $N_v(f)$ is the next flag in the cyclic order. Since $F=\amalg F_v$ one obtains a map $N:F\rightarrow F$. The orbits of the map $N \circ \imath$ are called the cycles 