User blog comment:B1mb0w/The Alpha Function/@comment-27173506-20160129134940/@comment-5529393-20160203102706

It seems actually quite reasonable that tree(3) = 2^18 - 4;  if you try to construct a sequence, you run out of options quite quickly, so taking the greedy strategy seems hard to beat. But I don't have good ideas on how to find upper bounds on tree(n) or any of these optimization problems;  there doesn't seem a way to be sure that a particular strategy is optimal. Of course, proving that tree(n) is a provably recursive function of some theory won't give us upper bounds for any particular value.

TREE(3) actually has stronger lower bounds; the strongest comes from this blog post

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