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Is this way to prove that all trees are bipartite graphs correct?

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Yep, this is a good way to prove that. I think you should also mention why every edge is between levels i and i+1; why can there not be an edge between levels i and i+2, or i+3?
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I don’t think this works since you are assuming there are even and odd levels to begin with (which aren’t even properly defined) and this is equivalent to being bipartite.

Instead you need to come up with some definition of “even” and “odd” and then prove that the intersection is empty. (Or do another proof)

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