I've written a recursive function of a tree, and I would like to know what it's called!  It's not quite the same as the height or the width of a tree, but it seems kind of like a width.Assuming the tree $T$ is binary, we can define the function $f(T)$ as follows.  If $T$ has a only a single node, let $f(T) = 1$.  Otherwise, let $T_0$ and $T_1$ be the two trees rooted at the child nodes $T$'s root.  If $f(T_0) = f(T_1)$, then let $F(T) = F(T_0) + 1$; otherwise, let $f(T) = \max(f(T_0), f(T_1))$.The function $f$ is generalized as follows to trees that are not necessarily binary.  As before, let $f(T) = 1$ when $T$ has no edges.  Otherwise, let $T_0, \dots, T_k$ be the $k+1$ subtrees rooted at the child nodes of $T$'s root, sorted by descending values of $f$.  Then let $f(T) = \max_i(f(T_i) + i)$.
What is the name of this function of a tree?
trees