Rooted Tree Corresponds to Arborescence

Theorem
Let $T = (V, E)$ be a rooted tree with root $r$.

Then there is a unique orientation of $T$ which is an $r$-arborescence.

Proof
Since a tree is connected and has no cycles, there is exactly one path from each vertex of $T$ to each other vertex of $T$.

Let $A$ be the set of all ordered pairs $x, y \in V$ such that:


 * $(x, y) \in E$ and
 * The unique path from $r$ to $y$ passes through $x$.