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  "body": "\n## Proof\n\n1. A [[tree|0MJ0JS72DZ]] $T$ is [[acyclic|0XNN9Z8E5A]], so it has no cycle at all.\n2. In particular, $T$ has no cycle of odd length.\n3. By the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]], $T$ is [[bipartite|07DVHJ8N0A]].\n\n## Idea\n\nThe only obstruction to splitting the vertices into two sides is an odd cycle. A tree has no cycles, so nothing can go wrong. Concretely, colour each vertex by the parity of its distance to a fixed root.\n",
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