## FANDOM

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We know the weak tree function has growth rate $$\vartheta(\Omega^\omega)$$ in FGH or HH. However, to find a good bound of TREE(3), it's not enough that we just know tree(n) is comparable to $$H_{\vartheta(\Omega^\omega)}(n)$$. We should know how it works. Here I found some results about tree function.

• Sequences whose length comparable to $$H_{\vartheta(\Omega^\omega)}(n)$$ (maybe not the winning sequences)
• A better bound for growth rate of TREE(n)
• A sequence of TREE(3) and a bound

We need a definition here for the second question.

## tree(n)

Sequences started with (()()()...()()) can have length comparable to $$H_{\vartheta(\Omega^\omega)}(n)$$. Now vertices have at most n children. Then it'll reduce to "all vertices have at most n-1 children", then "all vertices have at most n-2 children". Finally become a binary tree (but not ordered). It's at $$\varepsilon_0$$ level (in HH).

If we get (()()) and at most n vertices, the next tree is ((...(())...)) with n+1 vertices. At the end we get () and at most 2n+1 vertices, at the same time $$H_{\omega}(n)=2n$$, so we call (()()) has level $$\omega$$.

More comparisons:

tree level
(()()) $$\omega$$
((()())) $$\omega+1$$
(((()()))) $$\omega+2$$
((())()) $$\omega2$$
(((()))()) $$\omega3$$
((()())()) $$\omega^2$$
(((()())())) $$\omega^2+1$$
(((()()))()) $$\omega^2+\omega$$
(((())())()) $$\omega^22$$
((((()))())()) $$\omega^23$$
(((()())())()) $$\omega^3$$
((((()())())())()) $$\omega^4$$
((())(())) $$\omega^\omega$$
(((())(()))) $$\omega^\omega+1$$
(((())(()))()) $$\omega^\omega+\omega$$
((((())(())))()) $$\omega^\omega+\omega2$$
((((())(()))())()) $$\omega^\omega+\omega^2$$
(((()))(())) $$\omega^\omega2$$
((((())))(())) $$\omega^\omega3$$
((()())(())) $$\omega^{\omega+1}$$
(((()()))(())) $$\omega^{\omega+1}+\omega^\omega$$
(((())())(())) $$\omega^{\omega+1}2$$
(((()())())(())) $$\omega^{\omega+2}$$
((((()())())())(())) $$\omega^{\omega+3}$$
(((())(()))(())) $$\omega^{\omega2}$$
((((())(())))(())) $$\omega^{\omega2}+\omega^\omega$$
((((()))(()))(())) $$\omega^{\omega2}2$$
((((())(()))(()))(())) $$\omega^{\omega3}$$
(((()))((()))) $$\omega^{\omega^2}$$
((((())))((()))) $$\omega^{\omega^2}2$$
(((()()))((()))) $$\omega^{\omega^2+1}$$
(((())(()))((()))) $$\omega^{\omega^2+\omega}$$
((((()))((())))((()))) $$\omega^{\omega^22}$$
((((())))(((())))) $$\omega^{\omega^3}$$
((()())(()())) $$\omega^{\omega^\omega}$$
(((()()))(()())) $$\omega^{\omega^\omega}2$$
(((())())(()())) $$\omega^{\omega^\omega+1}$$
(((())(()))(()())) $$\omega^{\omega^\omega+\omega}$$
(((()())(()()))(()())) $$\omega^{\omega^\omega2}$$
(((()()))((()()))) $$\omega^{\omega^{\omega+1}}$$
((((()())))(((()())))) $$\omega^{\omega^{\omega+2}}$$
(((())())((())())) $$\omega^{\omega^{\omega2}}$$
((((()))())(((()))())) $$\omega^{\omega^{\omega3}}$$
(((()())())((()())())) $$\omega^{\omega^{\omega^2}}$$
((((()())()))(((()())()))) $$\omega^{\omega^{\omega^2+1}}$$
((((()()))())(((()()))())) $$\omega^{\omega^{\omega^2+\omega}}$$
((((())())())(((())())())) $$\omega^{\omega^{\omega^22}}$$
((((()())())())(((()())())())) $$\omega^{\omega^{\omega^3}}$$
(((())(()))((())(()))) $$\omega^{\omega^{\omega^\omega}}$$
((((()))(()))(((()))(()))) $$\omega^{\omega^{\omega^\omega2}}$$
((((())(()))(()))(((())(()))(()))) $$\omega^{\omega^{\omega^{\omega2}}}$$
((((()))((())))(((()))((())))) $$\omega^{\omega^{\omega^{\omega^2}}}$$
(((((())))(((()))))((((())))(((()))))) $$\omega^{\omega^{\omega^{\omega^3}}}$$
(((()())(()()))((()())(()()))) $$\omega^{\omega^{\omega^{\omega^\omega}}}$$
((((())(()))((())(())))(((())(()))((())(())))) $$\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}$$
((((()())(()()))((()())(()())))(((()())(()()))((()())(()())))) $$\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}$$

Then, the first tree that has 3-children vertices is (()()()), which has level $$\varepsilon_0$$. From $$\varepsilon_0$$ to $$\varepsilon_1$$ the (()()()) has no changes.

tree level
(()()()) $$\varepsilon_0$$
((()()())) $$\varepsilon_0+1$$
((()()())()) $$\varepsilon_0+\omega$$
((()()())(()())) $$\varepsilon_0+\omega^{\omega^\omega}$$
((()()())(()()())) $$\varepsilon_02$$
(((()()()))(()()())) $$\varepsilon_03$$
(((()()())())(()()())) $$\varepsilon_0\omega=\omega^{\varepsilon_0+1}$$
((((()()())()))(()()())) $$\omega^{\varepsilon_0+1}+\varepsilon_0$$
((((()()()))())(()()())) $$\omega^{\varepsilon_0+1}2$$
((((()()())())())(()()())) $$\omega^{\varepsilon_0+2}$$
(((()()())(()))(()()())) $$\omega^{\varepsilon_0+\omega}$$
(((()()())((())))(()()())) $$\omega^{\varepsilon_0+\omega^2}$$
(((()()())(()()))(()()())) $$\omega^{\varepsilon_0+\omega^\omega}$$
(((()()())(()()()))(()()())) $$\omega^{\varepsilon_02}$$
((((()()())(()()())))(()()())) $$\omega^{\varepsilon_02}+\varepsilon_0$$
((((()()()))(()()()))(()()())) $$\omega^{\varepsilon_02}2$$
((((()()())())(()()()))(()()())) $$\omega^{\varepsilon_02+1}$$
((((()()())(()()()))(()()()))(()()())) $$\omega^{\varepsilon_03}$$
(((((()()()))(()()()))(()()()))(()()())) $$\omega^{\varepsilon_03}2$$
(((((()()())(()()()))(()()()))(()()()))(()()())) $$\omega^{\varepsilon_04}$$
(((()()()))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}}$$
((((()()())))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}}2$$
(((()()())(()()()))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}$$
((((()()())(()()()))(()()()))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}2$$
((((()()()))((()()())))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}2}$$
(((((()()()))((()()())))((()()())))((()()()))) $$\omega^{\omega^{\varepsilon_0+1}3}$$
((((()()())))(((()()())))) $$\omega^{\omega^{\varepsilon_0+2}}$$
(((()()())())((()()())())) $$\omega^{\omega^{\varepsilon_0+\omega}}$$
((((()()())()))(((()()())()))) $$\omega^{\omega^{\varepsilon_0+\omega+1}}$$
((((()()()))())(((()()()))())) $$\omega^{\omega^{\varepsilon_0+\omega2}}$$
(((()()())(()()))((()()())(()()))) $$\omega^{\omega^{\varepsilon_0+\omega^{\omega^\omega}}}$$
(((()()())(()()()))((()()())(()()()))) $$\omega^{\omega^{\varepsilon_02}}$$
((((()()())(()()())))(((()()())(()()())))) $$\omega^{\omega^{\varepsilon_02+1}}$$
((((()()()))(()()()))(((()()()))(()()()))) $$\omega^{\omega^{\varepsilon_03}}$$
((((()()())(()()()))(()()()))(((()()())(()()()))(()()()))) $$\omega^{\omega^{\omega^{\varepsilon_02}}}$$
((((()()())(()()()))((()()())(()()())))(((()()())(()()()))((()()())(()()())))) $$\omega^{\omega^{\omega^{\omega^{\varepsilon_02}}}}$$

Next ((())()()) has level $$\varepsilon_1$$.

tree level
((())()()) $$\varepsilon_1$$
(((()))()()) $$\varepsilon_2$$
((()())()()) $$\varepsilon_\omega$$
(((()())(()()))()()) $$\varepsilon_{\omega^{\omega^\omega}}$$
((()()())()()) $$\varepsilon_{\varepsilon_0}$$
(((()()())(()()()))()()) $$\varepsilon_{\varepsilon_02}$$
(((())()())()()) $$\varepsilon_{\varepsilon_1}$$
(((()()())()())()()) $$\varepsilon_{\varepsilon_{\varepsilon_0}}$$
((())(())()) $$\zeta_0$$
(((())(())())()()) $$\varepsilon_{\zeta_0+1}$$
((((())(())()))()()) $$\varepsilon_{\zeta_0+2}$$
((((())(())())((())(())()))()()) $$\varepsilon_{\zeta_02}$$
((((())(())())()())()()) $$\varepsilon_{\varepsilon_{\zeta_0+1}}$$
(((()))(())()) $$\zeta_1$$
(((())(())())(())()) $$\zeta_{\zeta_0}$$
(((()))((()))()) $$\varphi(3,0)$$
((()())(()())()) $$\varphi(\omega,0)$$
((()()())(()()())()) $$\varphi(\varepsilon_0,0)$$
(((()()())(()()())())((()()())(()()())())()) $$\varphi(\varphi(\varepsilon_0,0),0)$$
((())(())(())) $$\Gamma_0$$
(((())(())(()))()()) $$\varepsilon_{\Gamma_0+1}$$
((((())(())(())))()()) $$\varepsilon_{\Gamma_0+2}$$
((((())(())(()))()())()()) $$\varepsilon_{\varepsilon_{\Gamma_0+1}}$$
(((())(())(()))(())()) $$\zeta_{\Gamma_0+1}$$
(((())(())(()))(()()())()) $$\varphi(\varepsilon_0,\Gamma_0+1)$$
(((())(())(()))((())(())(()))()) $$\varphi(\Gamma_0,1)$$
((((())(())(()))((())(())(()))())(()()())()) $$\varphi(\varepsilon_0,\varphi(\Gamma_0,1)+1)$$
((((())(())(())))((())(())(()))()) $$\varphi(\Gamma_0,2)$$
((((())(())(()))((())(())(())))((())(())(()))()) $$\varphi(\Gamma_0,\Gamma_0)$$
((((())(())(()))()())((())(())(()))()) $$\varphi(\Gamma_0,\varepsilon_{\Gamma_0+1})$$
((((())(())(()))((())(())(()))())((())(())(()))()) $$\varphi(\Gamma_0,\varphi(\Gamma_0,1))$$
((((())(())(())))(((())(())(())))()) $$\varphi(\Gamma_0+1,0)$$
((((())(())(()))()())(((())(())(()))()())()) $$\varphi(\varepsilon_{\Gamma_0+1},0)$$
((((())(())(()))((())(())(()))())(((())(())(()))((())(())(()))())()) $$\varphi(\varphi(\Gamma_0,1),0)$$
(((()))(())(())) $$\Gamma_1$$
(((())(())(()))(())(())) $$\Gamma_{\Gamma_0}$$
(((()))((()))(())) $$\varphi(1,1,0)$$
(((())(())(()))((())(())(()))(())) $$\varphi(1,\Gamma_0,0)$$
(((()))((()))((()))) $$\varphi(2,0,0)$$
((()()())(()()())(()()())) $$\varphi(\varepsilon_0,0,0)$$
(((()()())(()()())(()()()))((()()())(()()())(()()()))((()()())(()()())(()()()))) $$\varphi(\varphi(\varepsilon_0,0,0),0,0)$$
(()()()()) $$\varphi(1,0,0,0)$$
((()()()())()()) $$\varepsilon_{\varphi(1,0,0,0)+1}$$
((()()()())(())()) $$\zeta_{\varphi(1,0,0,0)+1}$$
((()()()())(()()()())()) $$\varphi(\varphi(1,0,0,0),1)$$
((()()()())(())(())) $$\Gamma_{\varphi(1,0,0,0)+1}$$
((()()()())(()()()())(()()()())) $$\varphi(\varphi(1,0,0,0),0,1)$$
((())()()()) $$\varphi(1,0,0,1)$$
((())(())()()) $$\varphi(1,0,1,0)$$
((())(())(())()) $$\varphi(1,1,0,0)$$
((())(())(())(())) $$\varphi(2,0,0,0)$$
((()()()())(()()()())(()()()())(()()()())) $$\varphi(\varphi(1,0,0,0),0,0,0)$$
(()()()()()) $$\varphi(1,0,0,0,0)$$
(()()()()()()) $$\varphi(1,0,0,0,0,0)$$

So that is a way up to SVO. Start with (()()...()()) then reduce them from bottom to top in those tables.

## TREE(m,n) and TREE(n)

The weak tree function tree(n)=TREE(1,n). But what about TREE(2,n), TREE(3,n) and more? Here I found a way to reduce them. Use (), [] and {} for different color of vertices.

A [] can reduce to (()()...()()) so it has level SVO. And more,

tree level
[] $$\vartheta(\Omega^\omega)$$
([]) $$\vartheta(\Omega^\omega)+1$$
(([])) $$\vartheta(\Omega^\omega)+2$$
([]()) $$\vartheta(\Omega^\omega)+\omega$$
([](()()()())) $$\vartheta(\Omega^\omega)+\varphi(1,0,0,0)$$
([][]) $$\vartheta(\Omega^\omega)2$$
(([][])(()()())) $$\vartheta(\Omega^\omega)2+\varepsilon_0$$
(([])[]) $$\vartheta(\Omega^\omega)3$$
(([]())[]) $$\omega^{\vartheta(\Omega^\omega)+1}$$
(([][])[]) $$\omega^{\vartheta(\Omega^\omega)2}$$
(([])([])) $$\omega^{\omega^{\vartheta(\Omega^\omega)+1}}$$
(([][])([][])) $$\omega^{\omega^{\vartheta(\Omega^\omega)2}}$$
([]()()) $$\varepsilon_{\vartheta(\Omega^\omega)+1}$$
([](())()) $$\zeta_{\vartheta(\Omega^\omega)+1}$$
([][]()) $$\varphi(\vartheta(\Omega^\omega),1)$$
([](())(())) $$\Gamma_{\vartheta(\Omega^\omega)+1}$$
([][][]) $$\varphi(\vartheta(\Omega^\omega),0,1)$$
([]()()()) $$\varphi(1,0,0,\vartheta(\Omega^\omega)+1)$$
([][][][]) $$\varphi(\vartheta(\Omega^\omega),0,0,1)$$
([][][][][]) $$\varphi(\vartheta(\Omega^\omega),0,0,0,1)$$
[()] $$\theta(\Omega^\omega,1)$$
[(())] $$\theta(\Omega^\omega,2)$$
[(()()())] $$\theta(\Omega^\omega,\varepsilon_0)$$
[[]] $$\theta(\Omega^\omega,\vartheta(\Omega^\omega))$$
[([])] $$\theta(\Omega^\omega,\vartheta(\Omega^\omega)+1)$$
[([][][])] $$\theta(\Omega^\omega,\varphi(\vartheta(\Omega^\omega),0,1))$$
[[()]] $$\theta(\Omega^\omega,\theta(\Omega^\omega,1))$$
[[[]]] $$\theta(\Omega^\omega,\theta(\Omega^\omega,\vartheta(\Omega^\omega)))$$
[()()] $$\vartheta(\Omega^\omega+1)$$
[[()()]] $$\theta(\Omega^\omega,\vartheta(\Omega^\omega+1)+1)$$
[([()()][()()][()()])] $$\theta(\Omega^\omega,\varepsilon_{\vartheta(\Omega^\omega+1)+1})$$
[[[()()]]] $$\theta(\Omega^\omega,\theta(\Omega^\omega,\vartheta(\Omega^\omega+1)+1))$$
[(())()] $$\theta(\Omega^\omega+1,1)$$
[[]()] $$\theta(\Omega^\omega+1,\vartheta(\Omega^\omega))$$
[[()()]()] $$\theta(\Omega^\omega+1,\vartheta(\Omega^\omega+1))$$
[[[()()]()]()] $$\theta(\Omega^\omega+1,\theta(\Omega^\omega+1,\vartheta(\Omega^\omega+1)))$$
[(())(())] $$\vartheta(\Omega^\omega+2)$$
[(()()())(()()())] $$\vartheta(\Omega^\omega+\varepsilon_0)$$
[[][]] $$\vartheta(\Omega^\omega+\vartheta(\Omega^\omega))$$
[[[][]][[][]]] $$\vartheta(\Omega^\omega+\vartheta(\Omega^\omega+\vartheta(\Omega^\omega)))$$
[()()()] $$\vartheta(\Omega^\omega+\Omega)$$
[(())()()] $$\vartheta(\Omega^\omega+\Omega,1)$$
[(())(())()] $$\vartheta(\Omega^\omega+\Omega+1)$$
[(())(())(())] $$\vartheta(\Omega^\omega+\Omega2)$$
[[][][]] $$\vartheta(\Omega^\omega+\Omega\vartheta(\Omega^\omega))$$
[()()()()] $$\vartheta(\Omega^\omega+\Omega^2)$$
[[][][][]] $$\vartheta(\Omega^\omega+\Omega^2\vartheta(\Omega^\omega))$$
[[][][][][]] $$\vartheta(\Omega^\omega+\Omega^3\vartheta(\Omega^\omega))$$
{} $$\vartheta(\Omega^\omega2)$$

So I get some bound for TREE(m,n) functions:

Growth rate of TREE(2,n)$$\geq\vartheta(\Omega^\omega2)$$

Growth rate of TREE(3,n)$$\geq\vartheta(\Omega^\omega3)$$

and so on.

TREE(n) go across all the TREE(m,n) for constant m, so growth rate of TREE(n)$$\geq\vartheta(\Omega^\omega\omega)$$.

However, those sequences are not the winning sequence, so I've no idea about the upper bound of TREE(n).

## TREE(3)

While evaluating TREE(3) (not TREE(n)), it's important to have a good beginning. After some comparisons, I found that sequence as follow. Some parts are not the same as the order in the tables in part 1 and 2.

(And thanks to Deedlit's idea, I get a new version.)

1. {}
2. [[]]
3. [()()]
4. [(()())] - Notice that [()()] isn't a minor of [(()())], and this can improve a lot.
5. [(((())))]
6. (([((()))]))
7. ([((()))][][])
8. ([((()))][]()()) - Not the same as in the table.
9. ([((()))]()()()())
10. ([((()))](())(())())
11. ([((()))](()()())()()) - Then reduce the (()()()) only. This will take tree(3) steps.

tree(3)+10. ([((()))]()()())

tree(3)+11. ([((()))][((()))](()tree(3)+1)) - Here the superscript means there're tree(3)+1 ()'s as children in a (()()...()()).

tree(tree(3)+1)+tree(3)+10. ([((()))][((()))]())

tree(tree(3)+1)+tree(3)+11. ([((()))][](()tree(tree(3)+1)+tree(3)+4))

tree(tree(3)+1)+tree(3)+12. ([((()))][]((())(())()tree(tree(3)+1)+tree(3)+1))

tree(tree(3)+1)+tree(3)+13. ([((()))][]((()()())()tree(tree(3)+1)+tree(3)+2))

tree(tree(3)+1)+tree(3)+14. ([((()))][](((())(()))()tree(tree(3)+1)+tree(3)+2))

tree(tree(3)+1)+tree(3)+15. ([((()))][]((((())())())()tree(tree(3)+1)+tree(3)+2))

tree(tree(3)+1)+tree(3)+16. ([((()))][](((((()())))())()tree(tree(3)+1)+tree(3)+2))

tree(tree(3)+1)+tree(3)+17. ([((()))][]((((((()()))())))()tree(tree(3)+1)+tree(3)+2))

tree(tree(3)+1)+tree(3)+18. ([((()))][]((((((()()))()))()tree(tree(3)+1)+tree(3)+2)()))

tree(tree(3)+1)+tree(3)+19. ([((()))][]((((((((()()))()))()tree(tree(3)+1)+tree(3)+2)))))

tree(tree(3)+1)+tree(3)+20. ([((()))][(())](((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+21. ([((()))](([()]))(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+22. ([((()))]([()][()])(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+23. ([((()))]([()][][][])(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+24. ([((()))]([()][][]()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+25. ([((()))]([()][]()()()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+26. ([((()))]([()]()()()()()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+27. ([((()))]([()](())(())()()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+tree(3)+28. ([((()))]([()](()()())()()()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2)))) - Again, the (()()()) takes tree(3) steps.

tree(tree(3)+1)+2tree(3)+27. ([((()))]([()]()()()()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+28. ([((()))]([()][](()tree(3)+4)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+29. ([((()))]([()][]((())(())()tree(3)+1)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+30. ([((()))]([()][]((()()())()tree(3)+2)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+31. ([((()))]([()][](((())(()))()tree(3)+2)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+32. ([((()))]([()][]((((())())())()tree(3)+2)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+33. ([((()))]([()][](((((()())))())()tree(3)+2)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+34. ([((()))]([()][]((((((()()))())))()tree(3)+2)()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+35. ([((()))]([()][]((((((()()))()))()tree(3)+2)())()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+36. ([((()))]([()][]((((((((()()))()))()tree(3)+2))))()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

tree(tree(3)+1)+2tree(3)+37. ([((()))]([()]([][])(((((((()()))()))()tree(3)+2)))()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2))))

Then, changes happen on the subtree ([][]), and this part will be

tree(tree(3)+1)+2tree(3)+38. ([]()())

tree(tree(3)+1)+2tree(3)+39. ([](()()))

tree(tree(3)+1)+2tree(3)+40. ([](((()))))

tree(tree(3)+1)+2tree(3)+41. (([]())((())))

tree(tree(3)+1)+2tree(3)+42. (((([])))((())))

tree(tree(3)+1)+2tree(3)+43. (((([]))((())))())

tree(tree(3)+1)+2tree(3)+44. (((((([]))((()))))))

tree(tree(3)+1)+2tree(3)+45. ((((([]))((())))))

tree(tree(3)+1)+2tree(3)+46. (((([]))((()))))

tree(tree(3)+1)+2tree(3)+47. ((([]))((())))

tree(tree(3)+1)+2tree(3)+48. ((((([])((())))())(()))(()))

tree(tree(3)+1)+2tree(3)+49. ((((((([])((()))))))(()))(()))

tree(tree(3)+1)+2tree(3)+50. ((((((([])((())))))(()))())(()))

tree(tree(3)+1)+2tree(3)+51. ((((((((([])((())))))(())))))(()))

tree(tree(3)+1)+2tree(3)+52. ((((((((([])((())))))(()))))(()))())

tree(tree(3)+1)+2tree(3)+53. ((((((((((([])((())))))(()))))(())))))

tree(tree(3)+1)+2tree(3)+54. (((((((((([])((())))))(()))))(()))))

tree(tree(3)+1)+2tree(3)+55. ((((((((([])((())))))(()))))(())))

tree(tree(3)+1)+2tree(3)+56. (((((((([])((())))))(()))))(()))

tree(tree(3)+1)+2tree(3)+57. ((((((((((([])((())))))(())))(()))())())())())

tree(tree(3)+1)+2tree(3)+58. ((((((((((((([])((())))))(())))(())))))())())())

tree(tree(3)+1)+2tree(3)+59. (((((((((((((([])((())))))(())))(()))))())))())())

tree(tree(3)+1)+2tree(3)+60. ((((((((((((((([])((())))))(())))(()))))()))())))())

tree(tree(3)+1)+2tree(3)+61. (((((((((((((((([])((())))))(())))(()))))()))()))())))

tree(tree(3)+1)+2tree(3)+62. ((((((((((((((([])((())))))(())))(()))))()))()))()))

tree(tree(3)+1)+2tree(3)+63. (((((((((((((([])((())))))(())))(()))))()))()))())

tree(tree(3)+1)+2tree(3)+64. ((((((((((((((((((([])((())))))(())))(()))))()))())())))))))

tree(tree(3)+1)+2tree(3)+70. ((((((((((((([])((())))))(())))(()))))()))())())

tree(tree(3)+1)+2tree(3)+71. (((((((((((((((((((((((((([])((())))))(())))(()))))())())))))))))))))))())

tree(tree(3)+1)+2tree(3)+72. ((((((((((((((((((((((((((([])((())))))(())))(()))))())()))))))))))))))())))

tree(tree(3)+1)+2tree(3)+74. ((((((((((((((((((((((((([])((())))))(())))(()))))())()))))))))))))))())

tree(tree(3)+1)+2tree(3)+75. (((((((((((((((((((((((((((((([])((())))))(())))(()))))())())))))))))))))())))))))

tree(tree(3)+1)+2tree(3)+81. (((((((((((((((((((((((([])((())))))(())))(()))))())())))))))))))))())

tree(tree(3)+1)+2tree(3)+96. ((((((((((((((((((((((([])((())))))(())))(()))))())()))))))))))))())

tree(tree(3)+1)+2tree(3)+127. (((((((((((((((((((((([])((())))))(())))(()))))())())))))))))))())

tree(tree(3)+1)+2tree(3)+65589. (((((((((((([])((())))))(())))(()))))())())())

Now I start using Hardy hierarchy. Notice that it need one step to "expand" so in that hierarchy it's actually $$H_\alpha(n)=H_{\alpha[n]}(n)+1>H_{\alpha[n]}(n)$$. So HH is actually slightly smaller.

$$>tree(tree(3)+1)+H_{\omega^3}(65534)$$. ((((((((((([])((())))))(())))(())))())())())

$$>tree(tree(3)+1)+H_{\omega^32}(65534)$$. (((((((((([])((())))))(())))(()))())())())

$$>tree(tree(3)+1)+H_{\omega^33}(65534)$$. ((((((([])((())))))(())))(()))

$$>tree(tree(3)+1)+H_{\omega^\omega+\omega^33}(65534)$$. (((((([])((())))))(()))(()))

$$>tree(tree(3)+1)+H_{\omega^{\omega2}+\omega^\omega+\omega^33}(65534)$$. ((((([])((()))))(()))(()))

$$>tree(tree(3)+1)+H_{\omega^{\omega2}2+\omega^\omega+\omega^33}(65534)$$. (((([])((())))(()))(()))

$$>tree(tree(3)+1)+H_{\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. (([])((())))

$$>tree(tree(3)+1)+H_{\omega^{\omega^2}+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. ([]((())))

$$>tree(tree(3)+1)+H_{\omega^{\omega^2}2+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. []

Now the whole tree becomes ([((()))]([()][](((((((()()))()))()tree(3)+2)))()())(((((((()()))()))()tree(tree(3)+1)+tree(3)+2)))). Then changes happen on the ([()][](((((((()()))()))()tree(3)+2)))()()). Notice that the subtree (((((((()()))()))()tree(3)+2))) has level $$\vartheta(\Omega^{tree(3)+1})\times(\omega^2+\omega+1)+2$$, and once it decreases one vertix the [] will change into ([][]...[][]) with as many []'s as possible - at level $$\theta(\Omega^\omega,1)$$. So

$$>H_{\theta(\Omega^\omega,1)\vartheta(\Omega^{tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,1)2+\omega^{\omega^2}2+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. ([()][]()()())

Then it reduce to ([()][][](()()...())) with as many ()'s as possible - at level $$\vartheta(\Omega^\omega)$$, so

$$>H_{\theta(\Omega^\omega,1)^2\vartheta(\Omega^\omega)+\theta(\Omega^\omega,1)\vartheta(\Omega^{tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,1)2+\omega^{\omega^2}2+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. [()]

Now the whole tree becomes ([((()))][()](((((((()()))()))()tree(tree(3)+1)+tree(3)+2)))). Notice that the subtree (((((((()()))()))()tree(tree(3)+1)+tree(3)+2))) has level $$\vartheta(\Omega^{tree(tree(3)+1)+tree(3)+1})\times(\omega^2+\omega+1)+2$$, and once it decreases one vertix the [()] will change into ([(())][][]...[][]) with as many []'s as possible - at level $$\theta(\Omega^\omega,3)$$, so

$$>H_{\theta(\Omega^\omega,3)\vartheta(\Omega^{tree(tree(3)+1)+tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,3)2+\theta(\Omega^\omega,1)^2\vartheta(\Omega^\omega)+\theta(\Omega^\omega,1)\vartheta(\Omega^{tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,1)2+\omega^{\omega^2}2+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$. ([((()))][()]())

At last the tree reduce to ([((()))][]()), ([((()))]()()), ([((()))]([(())][][][]...[][])), ([((()))]()), ([((()))]), [((()))], (). But those are only at level $$\theta(\Omega^\omega,3)2$$, which is nothing comparing to $$\theta(\Omega^\omega,3)\vartheta(\Omega^{tree(tree(3)+1)+tree(3)+1})\times(\omega^2+\omega+1)$$.

Finally, I get TREE(3)>$$H_{\theta(\Omega^\omega,3)\vartheta(\Omega^{tree(tree(3)+1)+tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,3)2+\theta(\Omega^\omega,1)^2\vartheta(\Omega^\omega)+\theta(\Omega^\omega,1)\vartheta(\Omega^{tree(3)+1})\times(\omega^2+\omega+1)+\theta(\Omega^\omega,1)2+\omega^{\omega^2}2+\omega^{\omega2}3+\omega^\omega+\omega^33}(65534)$$, and a more simple but slightly weaker bound is TREE(3)>$$H_{\theta(\Omega^\omega,3)\vartheta(\Omega^\omega)}(tree(tree(3)))$$.

Who has a better idea?