# Where does the function TREE take off? **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [November 3, 2024, 11:14pm UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636 "2024-11-03T23:14:14Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [November 3, 2024, 11:14pm UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/1 "2024-11-03T23:14:14Z")
TREE(0)= 0 (I presume), TREE(1)= 1, TREE(2)= 3, TREE(3)= ohmygod. So clearly a graphing of the function TREE would look like a nearly horizontal slope dead-ending against a vertical cliff. Somewhere between 2 and 3; but where? Is TREE still humanly cognizable at say 2.001? ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [November 3, 2024, 11:19pm UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/2 "2024-11-03T23:19:12Z")
there is no TREE(2.001). TREE is a game played with a certain amount of colours. You can’t have 2.0001 colours. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [November 3, 2024, 11:41pm UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/3 "2024-11-03T23:41:38Z")
He’s right that TREE(n) is only defined for integer n. But I don’t think he’s talking about the right thing > **[Kruskal's tree theorem](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem#TREE_function)** > > In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. > The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite r... ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [November 4, 2024, 12:10am UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/4 "2024-11-04T00:10:24Z")
> [@colinfred](#): > > there is no TREE(2.001). TREE is a game played with a certain amount of colours. You can’t have 2.0001 colours Although it may not be applicable in this case, I thought that there were any number of ways of extrapolating the output of functions when non-integer _N_ inputs are used. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [November 4, 2024, 2:35am UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/5 "2024-11-04T02:35:48Z")
> [@Lumpy](#): > > any number of ways There are. Any number being possibly zero or even an infinite number. You need to pick a meaningful one. None of this is a turn the handle process. You need insight into the nature of your series to create even one. TREE is so good-awful that one doubts any extension will be forthcoming before the sun goes cold. Then again, maybe some deep insight is lurking. One of the famous sets of extensions is of the Fibonacci series, which can be generalised in a few useful ways. > **[Generalizations of Fibonacci numbers](https://en.m.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Extension_to_all_real_or_complex_numbers)** > > In mathematics, the Fibonacci numbers form a sequence defined recursively by: > That is, after two starting values, each number is the sum of the two preceding numbers. > The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers. > Using > > > > > F > > n > − > ... The key being the nature of the recurrence relationship defining the sequence. TREE is not so amenable to such analysis. ---
**Author:** ![Measure\_for\_Measure](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/measure_for_measure/32/557_2.png) [@Measure\_for\_Measure](https://boards.straightdope.com/u/Measure_for_Measure)\ **Post date:** [November 4, 2024, 2:36am UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/6 "2024-11-04T02:36:38Z")
If you connected the values like they do in typical graphs, you would get a vertical line at TREE(2). To see this, consider the NotTree function with these values: NotTREE(1) = 1 NotTREE(2) = 3 NotTREE(3) = 1000 Zowie! Here are 2 graphs, done at different scales. [![](https://cdn.bsky.app/img/feed_fullsize/plain/did:plc:kem5cevl5sjpizk4s27cbyig/bafkreidi6uexmpl4cx4d2bboyvvsm6od2i2ssxgtv3rrpvye6kpuzrzoke@jpeg) ](https://cdn.bsky.app/img/feed_fullsize/plain/did:plc:kem5cevl5sjpizk4s27cbyig/bafkreidi6uexmpl4cx4d2bboyvvsm6od2i2ssxgtv3rrpvye6kpuzrzoke@jpeg) Connecting the values in this case may be somewhat misleading, but that’s what the chart would look like. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [November 4, 2024, 2:51am UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/7 "2024-11-04T02:51:53Z")
What we’re looking for here is an extension of the domain of the TREE function to the reals. I think a more appropriate example is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function), which is an analytic continuation of the factorial function. The factorial function is defined only on the natural numbers like TREE. The gamma function is defined on the reals (except the negative integers), but matches the factorial function on the naturals. But I don’t know if there’s any reasonable analytic continuation of TREE. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [November 4, 2024, 3:41am UTC](https://boards.straightdope.com/t/where-does-the-function-tree-take-off/1009636/8 "2024-11-04T03:41:33Z")
> [@markn\_1](#): > > What we’re looking for here is an extension of the domain of the TREE function to the reals. I think a more appropriate example is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function), which is an analytic continuation of the factorial function. The factorial function is defined only on the natural numbers like TREE. The gamma function is defined on the reals (except the negative integers), but matches the factorial function on the naturals. But I don’t know if there’s any reasonable analytic continuation of TREE Thank you, what I didn’t have the vocabulary to express.