or the boundaries of the graph. These sets have the induced cyclic order.

Notice that each boundary can be seen as a cyclic sequence of directed edges. The directions are as follows. Start with any flag $f$ in the orbit. In the geometric realization go along this half-edge starting from the vertex $\del(f)$, continue along the second half-edge $\imath(f)$ until you reach the vertex $\del(\imath(f))$ then continue starting along the flag $N(\imath(f))$ and repeat.

A tree with a cyclic order of the flags at each vertex is called planar. A planar tree has only one cycle $c_0$.

Planar planted trees
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A planted planar tree is a rooted planar tree $(\t,v_0)$ together with a linear order of the set of flags at $v_0$. Such a tree has a linear order of all flags as follows: Let $f$ be the smallest element of $\del^{-1}(v_0)$, then every flag appears in $c_0$ and defining the flag $f$ to be the smallest gives a linear order on the set of all flags. This linear order induces a linear order on all oriented edges and on all un-oriented edges, by restricting to the edges in the orientation opposite the natural orientation i.e. pointing away from 