Correction

Section 5 of this paper mentions an upper bound on the number of elementary trees rooted at a given node. That bound is so incorrect that I have no idea how I ended up with it. Thanks to Keith Hall for asking (on 2004-09-27; I replied the following day).

For example, if m=6, k=3, a given node is the root of at most 59 elementary trees, rather than at most 43 as claimed in the paper.

The correct bound is a generalization of the Catalan numbers. Fix m >= 1. There are precisely f(k) complete m-ary trees with exactly k internal (i.e., non-leaf) nodes, where f is defined by the recurrence

f(0) = 1
f(k) = g_m(k-1) for k > 0

where g_j is a helper function that defines a product of j terms:

g₁(k) = f(k)
g_j(k) = sum_i=0^k f(i) g_j-1(k-i) for j > 1

The familiar Catalan numbers are the case m=2, i.e., binary trees. (In that case, at least, f(k) has a closed form that is bounded above by 4^k. I haven't studied the question for larger m.)

Thus, a given node in an infinite, complete m-ary tree T is the root of exactly f(0)+f(1)+...+f(k) elementary trees with up to k internal nodes. (Only the internal nodes are freely determined; all children of an internal node must be included as frontier nodes.)

In our situation, T is a finite, not necessarily complete m-ary tree. However, this only reduces the number of elementary trees rooted at each node, so f(0)+f(1)+...+f(k) is still an upper bound. For the example of m=6, k=3, f(0)+f(1)+f(2)+f(3)=59.

Jason Eisner - jason@cs.jhu.edu (tech correspondence welcome)