# Directed Trees

In the following functions, the graph is assumed to be a directed tree. This means it is a tree containing a root vertex and all edges are directed away from that vertex.

## `IsRootedTree(G): GrphDir -> BoolElt, GrphVert`

Returns `true` exactly when the directed graph $G$ is a tree having a vertex $v$ such that all edges are directed away from $v$. In this case, the root vertex $v$ is returned as a second value.

## `Root(G): GrphDir -> GrphVert`

The root vertex of a rooted tree.

## `IsRoot(v): GrphVert -> BoolElt`

Returns `true` if and only if the graph containing the vertex $v$ is directed as a rooted tree with $v$ as root.

## `RootSide(v): GrphVert -> GrphVert`

When the graph containing the vertex $v$ is directed as a rooted tree, this returns the unique neighbouring vertex to $v$ which is closer to the root vertex. If $v$ is the root vertex, it is returned itself.

## `VertexPath(u, v): GrphVert, GrphVert -> SeqEnum`

A sequence of vertices comprising a path in a directed graph from the vertex $u$ to the vertex $v$. The path does not necessarily respect the edge directions. Indeed it will first trace back to a common ancestor of $u$ and $v$ and then follow edge directions to $v$.

## `BranchVertexPath(u, v): GrphVert, GrphVert -> SeqEnum`

The sequence of vertices on the vertex path from the vertex $u$ to the vertex $v$ having valency at least $3$.