Expected maximal degree of vertice in uniform spanning tree Inspired by this I am curious how to evaluate expected maximal vertices degree
in uniform spanning trees. I am not able to add comments, therefore I had to ask a new question. Misha

We can also use Prüfer codes to answer questions about the joint distribution of degrees of multiple vertices in the same way.

How can we do that?
 A: If we are picking a uniformly random labeled tree, this is the same as picking a uniformly random Prüfer code, and the maximum degree in the tree is one more than the maximum number of times that a label appears in the Prüfer code.
This is equivalent to the classic balls into bins problem: we are placing $n-2$ balls (the values of the Prüfer code) randomly into $n$ bins (the labels on the vertices) and we're looking for the maximum number of balls in a bin.
The Wikipedia article gives us a value of $(2+o(1))\frac{\log n}{\log \log n}$ for the maximum load when placing $n$ balls into $n$ bins. The answer with $n-2$ balls should be the same because the $2$ missing balls don't make a significant contribution asymptotically. (The $-1$ in the degree formula from the Prüfer code does not matter for the same reason.)
So with high probability, the maximum degree in the random tree is $(2+o(1))\frac{\log n}{\log \log n}$.
This also describes the expected value (which doesn't actually follow from the with high probability result, but it can be derived from the proof).
