I came to know that number of labelled unrooted trees on $n$ vertices is given by formula $n^{n-2}$. Also, I came to know that there is no closed formula for number of unlabelled graphs on $n$ vertices. I have following doubts:
- What is number of unlabelled unrooted trees?
- What is number of unlabelled rooted trees?
- What does it mean by tree in "labelled unrooted tree"? Does it mean graph connected with $n-1$ edges? Since I cannot imagine a tree without a root.
- Also I believe there is no such thing as labelled / unlabelled "unrooted m-ary" tree, since for being m-ary, there has to be some root. Am I correct with this? If no, then how we can have unrooted m-ary tree and what is count of such trees on $n$ vertices? (Number of unlabelled rooted m-ary tree is given by Cayley's number $\frac{1}{(m-1)n+1}\binom{mn}{n}\times n!$)