Consider the following recurrence:

$T(n)=3T(\frac{n}{3})+3T(\frac{n}{2})+n^2$

$T(1)=C$

We will show that $T(n)=O(n^{lo{g}_2(\frac{13}{3})}).$ To do this, start by examining the first three levels of the recursion tree, showing how to compute the amount of work at each level. From here, establish a formula for the amount of work on level $i.$ Then, determine the last level of the recursion tree $($note that it is sufficient to focus on the largest piece at level $i,$ as we are only concerned with a Big$-$O bound$).$ Finally, construct a summation which totals the amount of work over all levels and show why this summation is $T(n)=O(n^{lo{g}_2(\frac{13}{3})}).$