User blog comment:B1mb0w/Finite: a tour of the finite numbers/@comment-30754445-20170728042618

The lower bound you've given to TREE(n) is apperantly the actual growth rate of the function:

https://math.stackexchange.com/questions/1950116/where-does-treen-sit-on-the-fast-growing-hierarchy/1959090#1959090