Is a tree of CSPRNGs secure?
Yes, this is basically the Goldreich-Goldwasser-Micali construction of a PRF from a PRG. All leaves of the tree appear independently random, and furthermore the construction works even if the tree has exponentially many leaves (note that it is possible to compute the value at one leaf without computing the entire tree, by focusing just on the root-to-leaf path).

You can find a formal proof of security for this construction in section 6.3 of my book (PDF link).


  Goldreich, Oded, Shafi Goldwasser, and Silvio Micali. "How To Construct Random Functions." Foundations of Computer Science, 1984. 25th Annual Symposium on. IEEE, 1984.