I have a labelled tree with $n$ vertices for $n > 1$. How do I find the number trees with vertices tree that has a degree of $n-2$? I have been trying to figure it out but cannot seem to solve it. Is there any theorems that would help me with this?
I have tried using the Prufer sequence to solve it. I saw a pattern that if a vertex in the Prufer sequence shows up $n-2$ times, then that graph has a vertex with degree $n-2$. However, I am not sure how to work out the Prufer sequence for the larger number of vertices without drawing all the graphs which is not ideal.