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| = 1
- |T| = 2
- |T| = 4
- |T| = 14
- |T| > 243
- |T| > 2↑↑2295
- Friedman, Harvey. Enormous Integers in Real Life