If we consider labelled trees then this question has any easy solution: there are labelled trees on n vertices. Counting the total number of graphs on n vertices is even easier. However, no closed form solution is known for unlabelled trees and this problem has been open for well over a century.
Hmm, any node can be a root node which is tricky, but not show stopper. A root node can have 1 to N-1 branches not so bad. You can order them by some sort of weight with each leaf being (depth * 2N)^N so they are strictly ordered, still looking good. But, how do you count mirror images. AKA a simple star with a center point and N edges just counts as 1.
If we consider labelled trees then this question has any easy solution: there are labelled trees on n vertices. Counting the total number of graphs on n vertices is even easier. However, no closed form solution is known for unlabelled trees and this problem has been open for well over a century.
Hmm, any node can be a root node which is tricky, but not show stopper. A root node can have 1 to N-1 branches not so bad. You can order them by some sort of weight with each leaf being (depth * 2N)^N so they are strictly ordered, still looking good. But, how do you count mirror images. AKA a simple star with a center point and N edges just counts as 1.