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]| > 2
^{43} - |T[5]| > 2↑↑2
^{295}

## References

- Friedman, Harvey. Enormous Integers in Real Life

## See also

Main article: Harvey Friedman

Mythical tree problem · Friedman's vector reduction problem · **Friedman's finite ordered tree problem**· block subsequence theorem n(4) · Friedman's circle theorem · TREE sequence TREE(3) · subcubic graph number SCG(13) · transcendental integer · finite promise games · Friedman's finite trees · Greedy clique sequence

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