Friedman's finite ordered tree problem

The finite ordered tree problem was researched by Harvey Friedman.

Friedman defines an ordered tree as a triple (V,≤,<') where (V,≤) is a finite poset with a least element (root) in which the set of predecessors under ≤ of each vertex is linearly ordered by ≤, and where for each vertex, <' is a strict linear ordering on its immediate successors. He also defines the following:


 * Vertex x ≤* y if and only if x is to the left of y, or if x ≤ y.
 * d(v) is the position of v in counting from 1.

He then defines T[k] to be the tree of height k such that every vertex v of height ≤k - 1 has exactly d(v) children, and |T[k]| to be number of children.

Friedman has proven that |T[k]| has a similar growth rate to that of the Ackermann function. The first few values are as follows:


 * T[0]| = 1
 * T[1]| = 2
 * T[2]| = 4
 * T[3]| = 14
 * T[4]| > 243
 * T[5]| > 2↑↑22 95