Consider all labeled trees on vertices $\{1,\dots,n\}$. What is the average degree of vertex $1$?
I tried to count for each possible degree $i$ of the vertex $1$ the number of trees such that the vertex $1$ has exactly degree $i$. Then $i$ tried to compute the summation: $$S= \sum_{i=1,...,n-1} i *N_i $$ Where $N_i$ is the number of trees such that vertex 1 has degree i. I got $N_i$ = $ \binom{n-2}{i-1} (n-1)^{n-2-i}$ using prufer codes. The result at the end would simply be $\frac{S}{n^{n-2}}$. Is there a way to simplify the result ? The problem is the i in the summation, if there was not the $i$, I could simply use the binomial theorem. I also tried to take the derivative of $(1+x)^n$ but I could not conclude.
The average degree of vertex $1$ is one $n$'th of the sum of the average degrees of all the vertices, which is $2n-2$.
Therefore the average degree is $\frac{2n-2}{n} = 2 - \frac{2}{n}$