Note: this is from JeffE's Algorithms notes on Recurrences, page 5.
(1). So we define the recurrence $T(n) = \sqrt{n}T(\sqrt{n})+n$ without any base case. Now I understand that for most recurrences, since we're looking for asymptotic bounds, the base case wouldn't matter. But in this case, I don't even see where we could define the base case. Is there any number we are guaranteed to hit as we keep taking square roots starting from any integer Do we just define $T(n) = a$ for $n<b$, for some reals $a$, $b$?
(2). On page 7, Erickson gets that the number of layers in the recursion tree L will satisfy $n^{{2}^{-L}} = 2$. Where is this coming from? I have no idea. I see that the number of leaves in the tree should sum to $\sqrt(n)\sqrt(n) = n$, but I have no idea where to go from there.
Any help is appreciated!
Notes I'm looking at: http://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf