I have started reading some texts related to asymptotics of algorithms. Recently, I studied the recursion tree method and found some pretty interesting examples related to it that I could not solve. I don't know how to proceed with recursion trees in case we have a uneven divide and conquer recurrence in which the work at each level goes on increasing. For example, here is one problem I was stuck on.
$t(n) = 2t(n/3) + 2t(2n/3) + n$
Here at each level the cost of recursion is more than the previous level. So in this case the time complexity is decided by the root nodes. But how do I predict the number of root nodes of recursion tree without having n. And how do we approach such recurrences using recursion tree(without using master theorem or Akra bazzi theorem).
If possible can someone give me a generic solution for recurrences of the type :
$t(n)=\displaystyle\sum_{i=1}^{k}a_it(n/b_i)+f(n)$ where $b_i>1$
Using only recursion tree