In my combinatorics class, I have just met the following problem
Suppose we have a rooted tree (tree where one node is special as the root) with $n$ nodes. Further, we suppose that besides the root, the tree has no branching nodes) so we consider the following representative image
the blackened node is the root. Suppose we are given one of these trees, how can we count the number of different ways to label this tree (up to isomorphism of rooted trees with labels) with the $n$ letters $\{1,2,...,n\}$ such that a parent's label is always less than that of a child, so the following two lableings are allowed and actually count as the same
This problem has me stumped. I do not see a way, given one of these special rooted trees, to actually count the number of these special labelings. Is there a way, given one of these special rooted trees, to quickly write down the number of the different allowed labelings? I thank all helpers.