Suppose I have the labeled trees $T_{1}, \ldots, T_{n}$ with $b_{1}, \ldots, b_{n}$ vertices respectively. I would like to know how many ways I can compose a tree from these trees by using all trees? In this problem editorial they claim that there are $\prod_{i=1}^{n}{b_{i}} \cdot \left( \sum_{i = 1}^{n}{b_{i}} \right)^{n - 2}$ such compositions. I don't know how to prove this and there is no proof linked on the editorial either. Could someone help me with this?
2026-03-30 20:51:45.1774903905
How many ways can I connect labeled trees into a tree.
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The problem is listed in Lovasz's problem book. This is problem 4 in chapter 4. The original solution is attributed to J. W. Moon, but I haven't managed to find the article describing this. The proof is fairly complicated so it's not a surprise that it's omitted from the editorial.
EDIT: Seems like Igor Pak shared the original book as well, although the book by Lovasz is more accessible and the proof is complicated but well presented.