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We can use Prüfer sequences (of length $n-2$) to find the labeled spanning trees for $K_n$, using the following decoding algorithm, by Cayley’s theorem the number of spanning trees are $n^{n-2}$.
 

For $n=5$, there are $5^3=125$ such spanning trees on $5$ labeled vertices, as can be computed using the above algorithm
and seen from the following animation:

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