I've been trying to write up a proof recently which involves several properties of unlabeled rooted trees. Do any of these properties have commonly used names I am not aware of (so that I need not define them in my proof), or are any of the names I have chosen used for other properties of trees (so that I should choose a different name to avoid confusion)?
An exhaustive list of terms I'm using, to be on the safe side:
Given an unlabeled rooted tree, I say that a node in the tree has depth equal to the distance between it and the root and the tree has depth equal to the maximum depth among all non-leaf roots.
A node is maximal if its depth equals that of the tree and no other node at that depth has greater degree. The number of leaves adjacent to the maximal node in a tree (0 if no such node exists) is the deepest degree of the tree.
The subtree descending from each child of the root is a branch of the tree, and a branch containing a maximal node of the tree is a maximal branch.
Finally, a branch is undecorated if every leaf in the branch is at the same depth.