User blog comment:Hyp cos/tree function and TREE(3)/@comment-5529393-20140625064232

I believe the following is an improvement:

9. ([(())])

10. ([(())]()())

11. ([(())]())

We can now work on reducing the (). We know that tree(3) >= 262140, so we can have a sequence of at least 262140 trees ending in:

262150. ([(())])

262151. ([(())][(())](...))  {262141 's}

I'm not totally certain of the best way to continue from here, but we can certainly just reduce the (...), for more than $$H_{\vartheta(\Omega^262139)}(262141)$$ steps.

$$ > H_{\vartheta(\Omega^262139)}(262141)$$. ([(())][](...))

So we get TREE(3) $$ > H_{\theta(\Omega^{\omega}, 3) \vartheta (\Omega^c)} (262141)$$, where $$ c = H_{\vartheta(\Omega^262139)}(262141)$$.