appear in the story. In the sequel we will mainly follow the notations of [@GeomInt].

1.  [A **graph** $t$ is given by a set of vertices $V_t=\{1,\dots,n\}$ and a set of edges $E_t$ which is a set of pairs of elements of $V_t$. We denote the number of vertices by $|t|$. An **isomorphism** between two graphs $t$ and $t'$ having the same number of vertices is a permutation $\sigma\in S_{|t|}$ such that $\{\sigma(v),\sigma(w)\}\in E_{t'}$ if $\{v,w\}\in E_{t}$. Two graphs are called **equivalent** if there is an isomorphism between them. The **symmetries** of a graph are the automorphisms of the graph. We denote the group of symmetries by $sym(t)$.]{}

2.  [A **tree** is a graph which has no cycles. Isomorphisms and symmetries are defined the same way as for graphs ]{}

3.  [A **rooted tree** is a tree with one distinguished vertex. An **isomorphism** of rooted trees is an isomorphism of graphs which sends the root to the root. Symmetries and equivalence are defined correspondingly.]{}

4.  [A **bipartite graph** is a graph $t$ together with a map $\omega:V_t\rightarrow
                            \{\circ,\bullet\}$ such that $\omega(v)\neq\omega(w)$ if $\{v,w\}\in E_t$. An isomorphism of bipartite trees is an isomorphism of graphs which respects the coloring, i.e., $\omega(\sigma
                            