yields a one-dimensional CW complex whose realization we call the realization of the graph.

### Trees

A graph is connected if its realization is. A graph is a tree if it is connected and its realization is contractible.

A rooted tree is a pair $(\t,v_0)$ where $\t$ is a tree and $v_0\in V_{\t}$ is a distinguished vertex. In a rooted tree there is a natural orientation for edges, in which the edge points toward the root. That is we say $(f,\imath (f))$ is naturally oriented if $\del(\imath(f))$ is on the unique shortest path from $\del(f)$ to the root. This means that the set $E(v)$ splits up into incoming and outgoing edges. Given a vertex $v$, we let $|v|$ be the number of incoming edges and call it the arity of $v$. A vertex $v$ is called a leaf if $|v|=0$. Notice that the root is the only vertex for which $|v_0|=\val(v_0)$. For all other vertices $v\neq v_0$ one has $|v|=\val(v)-1$.

A bi-colored or black and white (b/w) tree is a tree $\t$ together with a map $\color:V\rightarrow \mathbb{Z}/2\mathbb{Z}$. Such a tree is called bipartite if for all $f\in
F_{\t}:\color(\del(f))+\color(\del(\imath(f)))=1$, that is edges are only between black and white vertices. We 