Talk:TREE sequence/Archive 1

Can we get the original source for this instead of Wikipedia? FB100Z &bull; talk &bull; contribs 02:18, September 13, 2012 (UTC)

Perhaps it comes from "Enormous Integers In Real Life". Ikosarakt1 (talk) 15:20, October 6, 2012 (UTC)

Anyways, what is the TREE sequence anyway? The article doesn't explain it. FB100Z &bull; talk &bull; contribs 03:54, October 20, 2012 (UTC)

See "chapter 11" for formal definition. As for values, see: http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html Ikosarakt1 (talk) 08:00, October 20, 2012 (UTC)

Comparison of TREE with array notations
I guess that TREE(3) is between humongulus and golapulus. --Cloudy176 (talk) 13:57, October 24, 2012 (UTC)

In fact: TREE(3) larger than fГ(3) = {3,3,1,2} & 3, which is larger than humongulus = {10,10,100} & 10. Ikosarakt1 (talk) 18:57, October 25, 2012 (UTC)

Hmm, my reasoning leads me to believe fГ 0 (n) is about n & n & n... & n with n n's. This would make TREE(3) much larger than golapulus Deedlit11 (talk) 16:19, November 16, 2012 (UTC)

Golapulus is already ~ {10,100 [1 [1 --| 1 [2] 2] 2} (there: --| is negation sign), but TREE(n) growth is about {3,n [1 [1 --| n] 2] 2} (small Veblen ordinal) Ikosarakt1 (talk) 09:07, November 17, 2012 (UTC)

I assume you are getting those correspondences from Bird's "Bowers Named Numbers", is that correct?? I'm not sure I agree with all the findings.? As I explained in my blog comment, I think Gamma_0 should correspond to n & n & n ... & n with n n's.? So the small Veblen ordinal will certainly be greater than that.

I should point out that the small Veblen ordinal is merely a lower bound for the growth rate of TREE(n). As far as I know, no good upper bound has been found.Deedlit11 (talk) 00:57, November 23, 2012 (UTC)

In this case, need to reconsider the comparison of TREE with array notations. Ikosarakt1 (talk) 12:16, November 23, 2012 (UTC)

After more detailed research on fast-growing hierarchy I am sure that TREE(3) far larger than meameamealokkapoowa oompa (see blog comment). Ikosarakt1 (talk) 15:33, November 24, 2012 (UTC)

TREE(3) made into a game --Cloudy176 (talk) 10:25, December 26, 2012 (UTC)

If your calculations are all correct, then (say) TREE(1,000) will certainly be larger than meameamealokkapoowa oompa. However, I still have doubts that TREE(3) is larger than meameamealokkapoowa oompa. --I want more clouds! 05:02, December 31, 2012 (UTC)

\(f_{\psi(\Omega^\omega)}\)(n)
Is \(\psi(\Omega^\omega)\) is the same ordinal as \(\vartheta(\Omega^\omega)\)? Ikosarakt1 (talk) 10:45, February 23, 2013 (UTC)


 * No, they're different. \(\psi(\Omega^\omega)=\vartheta(\omega)\), while \(\vartheta(\Omega^\omega)=\psi(\Omega^{\Omega^\omega})\). See . I want more clouds! 11:29, February 23, 2013 (UTC)
 * Hmm, ordinal collapsing function definitions may vary by authors. Maybe they are equal, or may not... I want more clouds! 11:31, February 23, 2013 (UTC)

Cloudy is correct, in that under the most usual definitions,  \(\vartheta(\Omega^\omega)=\psi(\Omega^{\Omega^\omega})\). Both these notations represent the small Veblen ordinal. Under the most usual definition,  \(\psi(\Omega^\omega)\) = \(\phi(\omega, 0)\), not the small Veblen ordinal. Therefore I will fix the main page.Deedlit11 (talk) 14:40, February 28, 2013 (UTC)

Ordered vs unordered siblings
I wanted to point out that there are two definitions of homeomorphic embeddability with rooted trees: the one with ordered siblings, and other one with unordered. On this site and on Goucher's blog we are with no doubt using unordered version, for which trees ([]) and ([]) are equivalent. But, if I'm not mistaken, even Kruskal himself used ordered version. This is because he interpreted trees as set of integer strings closed under prefixes. Friedman certainly used this version, because otherwise n(4) argument fails. Here is same story. Ordered siblings make much difference in tree(n) function too. Let utree(n) mean unordered version and otree(n) ordered version. It should be clear that \(utree(n)\leq otree(n)\). I found out that \(utree(2)=otree(2)=5\), \(utree(3)\geq 2^{17}+8\) and \(otree(3)\geq 2\uparrow ^6 3\), so difference quickly becomes significant. Which version you think we should use? Friedman's one, or Goucher's one? LittlePeng9 (talk) 12:43, February 23, 2013 (UTC)

Like busy beaver, TREE function also allows variations. Whatsoever, it is doesn't cancel Goucher's lower bound for TREE(3). Ikosarakt1 (talk) 13:22, February 26, 2013 (UTC)

Yes, you are right. Even better - Goucher's bound was given for unordered trees (I think) so, as ordered embeddability implies unordered one but not vice versa, this lower bound still applies when using utree function, but also otree function. And I'm sure we could add at least one more exponent to this lower bound. LittlePeng9 (talk) 14:15, February 26, 2013 (UTC)

Actually, both Friedman's posts on the FOM mailing list, and my answer on MathOverflow, use unordered trees (Friedman calls ordered trees structured trees.)  In fact, I made a point of it in my answer on MathOverflow how my argument had to be changed slightly because I was using unordered trees instead of the ordered trees used by Jervell. Please see the Friedman's FOM postings for an explanation of how n(4) can be embedded in an unordered tree.Deedlit11 (talk) 14:36, February 28, 2013 (UTC)

Ah, sorry, I haven't noticed how you modified Levitz's ordering. Now it makes sense to me. And about Friedman's posting, I've read it, but when I read about linearly ordered vertices starting at the root, what I thought it means is ordered tree with root and vertices attached directly to it. Thanks for all the clarification. LittlePeng9 (talk) 17:03, February 28, 2013 (UTC)

Bird's hierarchy
When Chris Bird writes that"[X] has level \(\alpha\)", he doesn't means that \(f_{\alpha}(n)\) is about {n,n [X] 2}. Rather, he creates his own hierarchy of separators, if you look closer. Ikosarakt1 (talk) 10:04, February 26, 2013 (UTC)


 * But that is irrelevant to the information you removed. That was the result Chris Bird has shown in the paper. I don't think it should be removed. I want more clouds! 10:55, February 26, 2013 (UTC)

Where Bird was stated that? I opened "Beyond Nested Arrays II" document, page 10, but Bird doesn't says there about TREE(3) strictly. Ikosarakt1 (talk) 11:10, February 26, 2013 (UTC)

You seem older version of his work. in the new version  "Beyond Nested Arrays II"  of 19.02.2013 on page 10. Konkhra (talk) 23:20, February 26, 2013 (UTC)


 * Here's the link: -- I want more clouds! 11:29, February 26, 2013 (UTC)

Okay, it turns out that. I will return that info. Thanks. Ikosarakt1 (talk) 12:05, February 26, 2013 (UTC)

Reasonable amount
From the article: "Actually, it has been proven [3] that TREE (3)> n (n (... (5) ...)) with any reasonable amount of nested functions". How much "reasonable amount"? thousand, googol or can be {3,3,3,3,3} in BEAF? Konkhra (talk) 22:20, February 27, 2013 (UTC)

I know it isn't well stated. It's actually larger than anything you may suspect - it's well larger than {3,3,...,3}, where there are {3,3,...,3) 3's, where there are {3,3,...,3) 3's... You can repeat this millions of billions of trillions times, so this is above what I'd call "reasonability limit". Here you can see deriviation of much stronger bound (in every place function F is used, function n will do)  LittlePeng9 (talk) 06:25, February 28, 2013 (UTC)


 * It's great! thanks for the answer Konkhra (talk) 08:07, February 28, 2013 (UTC)

Larger lower bound for TREE(3).
I have found a lower bound for TREE(3) that is larger than AP Goucher's. I wrote up an explanation of this in an answer on mathstackexchange, and I reproduce the explanation here.

Before we get to TREE(3), let's examine some smaller sequences that will build up towards it.

To describe a tree, I will use to denote a vertex labelled with 1, and [] to denote a vertex labelled with 2. The children of a vertex will be placed within the separators for the vertex; so for example ([][][]) means a vertex labelled with 1 with three children labelled with 2; and [() ] means a vertex labelled with 2 with two children labelled with 1; the left child has two children labelled with 1.

We will start by examining trees that are paths, with the root labelled with 2 and the rest of the vertices labelled with 1. [] () starting from a single vertex labelled with 2 leads to a sequence of length 3. [] ([]) ((([]))) (([])) ([]) [] () the last tree is the first tree in the sequence for tree(7) so we get a sequence of length greater than tree(7) [()] ([]) ((([]))) (([])) ([]) [] ([]) the last tree is the first tree in the sequence for tree(8), except the last vertex is labelled with a 2. We can thus continue with a sequence of length tree(8) of trees with all but one vertex labelled with 1, finally ending in a tree consisting of one vertex labelled with 2. The next tree is then a tree with more than tree(8) vertices with all vertices labelled with one;  this leads to a sequence of more than tree(tree(8)) vertices.

Continuing in this fashion, a path with one vertex labelled with 2 and n vertices labelled with 1 will lead to a sequence of more than tree\(^n(n+6)\) trees. If we define tree\(_2 (n)\) to be tree\(^n(n)\), then our lower bound is more than tree\(_2 (n)\) trees.

Now consider a tree consisting of a path of length \(n+1\) with the bottom vertex of the path having two children. Again, the root will have label 2 and the rest of the vertices will have label 1. For example, with n = 3 the tree is [((()))] We can construct a sequence of more than tree\(_2 (n-1)\) trees by basically following the previous sequence, with the two children at the bottom added on. This leads us to the tree []. We follow that with the tree [(((......)))] with more than tree\(_2 (n-1)\) vertices. By our previous bound, we will wind up with a sequence of length greater than tree\(_2\) (tree\(_2 (n-1))\).

If we next consider a tree similar to the previous one, except we add a child to one of the two children at the bottom, we will get a lower bound of tree\(_2\) (tree\(_2\) (tree\(_2 (n-1)))\). If we add a path of length $m$ rather than a single child, we get a lower bound of tree\(_2^{m+2}(n-1)\). Define tree\(_3 (n)\) to be tree\(_2^n(n)\). If \(n \ge m+3\), we have a lower bound of tree\(_3 (m)\).

Now we are ready to find a lower bound for TREE(3). Start with: {}  (one vertex with label 3) ([][]) [] [()()] [(() ) ] [((())) ] [(((())))] ([((()))]) ((([((()))]))) (([((()))])) ([((()))]) [((()))] ([(())]) this leads to a sequence of tree(8) trees, ending in [(())] this is followed by [(())] where the ( stands for tree(8) ('s and ) stands for tree(8) )'s.

This leads to a sequence of tree\(_2\) (tree(8)) trees, ending in [()] This is followed by [((  ))] where the ( stands for tree\(_2\) (tree(8)) ('s and ) stands for tree\(_2\) (tree(8)) )'s.

This leads to a sequence of more than tree\(_3\) (tree\(_2\) (tree(8))) trees.

Thus TREE(3) > tree\(_3\) (tree\(_2\) (tree(8))).

As you can imagine, the TREE(n) function clearly outpaces the tree(n) function, which is already at the level of the Small Veblen Ordinal in the fast-growing hierarchy. This is not surprising, since labelled trees lead to more possibilities than unlabelled trees.

In comparison to Bird's array notation, I believe that your bound should be about \(\{3,\{3,\{3,7 [1 [1 \neg 1,2] 2] 2\},2 [1 [1 \neg 1,2] 2] 2\},3 [1 [1 \neg 1,2] 2] 2\}\) Ikosarakt1 (talk) 22:30, March 6, 2013 (UTC)

Thanks. Unfortunately, I know of no firm upper bounds for either tree(n) or TREE(n). Deedlit11 (talk) 23:19, March 6, 2013 (UTC)

If I understand you correctly that tree\(_3\) (tree\(_2\) (tree(8))) = treetree tree(8)

but it is much smaller than the lower bound  Goucher's. may be I misunderstood? Konkhra (talk) 09:51, March 7, 2013 (UTC)

No, I believe he mean \(tree_2(n)\) is treetree tree ... tree(n)... (n) (n) (n) (with n occurrences of tree(n)), and \(tree_3(n)\) is tree2tree2 tree2 ... tree2(n)... (n) (n) (n) (with n occurrences of tree2(n)). Ikosarakt1 (talk) 09:59, March 7, 2013 (UTC)

Thanks, now I see the light Konkhra (talk) 10:05, March 7, 2013 (UTC)

Actually, \(tree_2(n)=tree^n(n)\) and \(tree_3(n)=tree_2^n(n)>tree^{tree^{tree^{...}(n)}(n)}(n)\) with n exponents.

LittlePeng9 is correct. AP Goucher's bound is therefore about \(tree_3(5)\). \(tree_3 (tree_2 (tree(8) ) ) \) is about treetree tree ... tree(n)... (n) (n) (n) with n exponents, where \(n = tree_2 (tree(8)) = tree^{tree(8)} (tree(8)) \). So my bound is not fundamentally larger than Goucher's bound, but it's an improvement nonetheless. Deedlit11 (talk) 00:18, March 8, 2013 (UTC)

By the way, thanks Cloudy for tidying up my post. Deedlit11 (talk) 00:57, March 12, 2013 (UTC)

Homeomorphic embedding
Given two trees: [[]] and [([])]. Is there a place for the homeomorphic embedding (1st to 2nd or otherwise) or not? It seems that I almost understood how that function really works. Ikosarakt1 (talk ^ contribs) 19:56, March 15, 2013 (UTC)

There's no homeomorphic embedding either way. To embed the first into the second, the four edges of the first must map to four disjoint paths (the "homeomorphic" part implies that different edges must map to disjoint paths), and the second does not contain four disjoint paths. To embed the second into the first, the path of length 2 must map to a path of length 2 or more, and the first doesn't have any. Deedlit11 (talk) 20:39, March 15, 2013 (UTC)

Okay, thanks for the explanation. Ikosarakt1 (talk ^ contribs) 20:49, March 15, 2013 (UTC)

I think explantation in article is wrong. It's more sophiscated than substring. If it were so, we'd have infinite sequence: ; ([]); ; ([]) etc. There is allowable modification - we can delete pair of neighbouring parenthesis, e.g. is embeddable to ([]). In unordered version we can interchange siblings. Optimal sequence for TREE(2) is []; (); LittlePeng9 (talk) 20:39, March 16, 2013 (UTC)

LittlePeng9 is correct. For example, {} is homeomorphically embeddable into the fourth example string given in the article, since {} is just a single vertex labelled with 3, so it is embeddable into any tree with a vertex labelled 3, i.e. any string containing braces. Deedlit11 (talk) 21:51, March 16, 2013 (UTC)

Exact values of tree(n)
Anyone knows exact values for \(tree(n)\) for specific n? I found that \(tree(1) \geq 2\), using the sequence \((), \). Also, \(tree(2) \geq 5\), using the sequence \((), ((())), (()), (), \). Ikosarakt1 (talk ^ contribs) 22:30, March 16, 2013 (UTC)

It is indeed true that tree(1) = 2 and tree(2) = 5. To prove that tree(2) = 5:  Define an n-path to be the tree consisting of a path of n edges starting from the root, and an n-star to be the tree where the root has n children, none of which have children. The first tree must be either an 2-path or an 2-star (or a subtree, and it is never better to take a subtree over a larger tree). If it is a 2-path, then there can be no more paths of length two or more in following trees, so the remaining trees can only be n-stars. So the best choices for the remaining trees must be to take the largest possible n-star each time, so you get (()), (), (), (),. Otherwise, the first tree is a 2-star, which means the remaining trees can have no vertices with two or more children. So the remaining trees must be n-paths, and again the best thing to do is to take the largest n-path each time, so you get (), ((())), (()), (),. These are the two longest sequences, and so tree(2) = 5.

For tree(3), I got a sequence of length 2^18 - 4;  LittlePeng9 got a sequence of length 2^17 + 8. I think it is reasonably possible that tree(3) = 2^18 - 4, but it will be very difficult to prove.

For tree(4) and above, the numbers are very large. For example, tree(4) will be larger than F_epsilon_0 (n) for very large n (larger than Graham's number to take a random large number). tree(5) will be larger than F_phi(1,0,0,0) (n) and so on. Deedlit11 (talk) 22:47, March 16, 2013 (UTC)

Chris Bird  wrote in his work "Beyond Nested Arrays II":

"TREE(3) > tree^(tree^(tree^(tree^(tree8 (7)) (7)) (7)) (7)) (7),

where the slightly slower growing tree function (lower case) grows at least as quickly as

f(n) = {3, n [1 [1¬1,2] 2] 2}

which is at the level of the small Veblen ordinal (much greater than Γ0). Royce Peng has investigated

that the growth rate of the slower tree function is at the θ(Ω^(n-2)) level for n ≥ 4, meaning that

tree(n) > {3, 3 [1 [1 ¬ n] 2] 2} (for n ≥ 4)."Konkhra (talk) 08:17, March 17, 2013 (UTC)