the root. We denote the latter by $\prec$ and its restriction to $E(v)$ or $F(v)$ by $\prec_v$.

We will equivalently consider planar planted trees as defined above or as a rooted planar trees whose root vertex has valence one. The bijection in one direction is given by adding a new root vertex and one new edge such that the induced linear structure on the old root is the given one. This tree is called the realization of the planar planted tree. In the other direction the bijection is simply given by contracting the unique edge incident to the root, but retaining the linear order. In the realization of a planar planted tree, we call the unique edge incident to the (new) root $v_{root}$ the root edge and denote it by $e_{root}$ and set $f_{root}$ to be the flag of the root edge which is not incident to the root. Also $E_{root}=\{e_{root}\}, F_{root}=\{f_{root}\}$.

An angle at a vertex $v$ in a planar tree is a pair of two flags incident to $v$ of which one is the immediate successor of the other in the cyclic order of $F_v$. There is a bijection between angles, flags and edges by associating to an 