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]| = 82*238-2
*|T[5]| > 2↑↑2295
== References ==
* Friedman, Harvey. Enormous Integers in Real Life
== See also ==
Category:Functions
Category:Theorems