Final answer:
To solve the given recurrence relation t(n) = t(n/3) * t(2n/3) * O(n) using the recurrence tree method, we can calculate the time complexity by creating a recurrence tree and summing up the work done at each level. The time complexity is O(n * log base 9 of n).
Step-by-step explanation:
To solve the recurrence relation t(n) = t(n/3) * t(2n/3) * O(n) using the recurrence tree method, we can visualize the problem by creating a recurrence tree. Let's start by expanding the equation:
t(n) = t(n/3) * t(2n/3) * O(n)
t(n) = [t(n/9) * t(2n/9) * O(n/3)][t(2n/9) * t(4n/9) * O(2n/3)] * O(n)
Continuing this process, we can see that the recurrence tree will have a depth of log base 9 of n. We can then calculate the total time complexity of the recurrence relation by summing up the work done at each level of the tree. This will give us the time complexity of O(n * log base 9 of n) for the given recurrence relation.