Problem with Understanding a Recursion Tree

Question

Consider the recursion tree:

$T(p) = 3T(\frac{2p}{8}) + 2T(\frac{p}{8}) + O(p)$.

I determined that there are at most $1 + log_{4}\ p$ levels, because the longest simple path from root to leaf is $p \rightarrow \frac{2p}{8} \rightarrow \frac{4p}{16} \rightarrow \frac{8p}{32} \rightarrow\ ...$.

This means that the time complexity is $O(p\ log\ p)$.

Now, I stopped all my leaves in the above recursion tree have $T(1)$, but say I have the condition $T(0) + O(1)$. Does this change my solution somehow?

Answer 1

Think of it this way, you have a mathematically formal description of a recursive function. Which every function maps an element (or more than one element) To 1 element in the codomain.

What would T(0) be in terms of this functions as an input? It wouldn't change the time complexity.