Is there existing terminology for the following property of a rooted tree?
I'd like to know how many choices of root node can produce an isomorphic tree. This is different from the number of isomorphic trees. For example, consider this tree.
If the node labeled '2' is the root, there are 4 total choices of root that produce isomorphic trees (2, 10, 8, and 15).
If 3 is the root, then there are 2 total choices that would produce isomorphic trees (3 or 12).
If 9 is the root, there is only one choice (9).
Is there a name, either for (a) this property of equivalence between roots, or (b) the number of nodes sharing this property?