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How to prove that in any tree, all longest paths cross one another in one vertex?

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    $\begingroup$ Did you already try to build up a proof for the claim but got stuck somewhere? For instance, did you try to do a proof by contradiction? $\endgroup$ – Juho Jun 10 at 10:03
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You can prove that any longest path contains the center vertices of the tree. It will then follow that all longest paths intersect not only at some vertex, but at the center.

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