# Rooted Tree Corresponds to Arborescence

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## Theorem

Let $T = \struct {V, E}$ be a rooted tree with root $r$.

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

## Proof

Recall that a tree is connected and has no cycles.

Thus there is exactly one path from each node of $T$ to each other node of $T$.

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

- $\tuple {x, y} \in E$ and
- The unique path from $r$ to $y$ passes through $x$.

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## Sources

- 1968: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*: $\S 2.3.4.2$