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+# Review 4-2
+
+* Hajin Ju, 2024062806
+
+## Problem 1
+
+Guess the solution to the recurrence $T(n) = T(n/3) + T(2n/3) + cn$, where $c$ is constant, is $\Theta(n\lg n) by applealing to a recursion tree.$
+
+### Solution 1
+
+* level 0
+$\Sigma = cn$
+* level 1
+$\Sigma = cn/3 + 2cn/3 = cn$
+* level 2
+$\Sigma = cn/9 + 2cn/9 + 2cn/9 + 4cn/9 = cn$
+* level k
+$\Sigma = cn$
+* level h
+$\Sigma = cn$
+
+The shortest depth $n\to 1$ is $h = \lg_{3/2}{n}$
+The longest depth $n\to 1$ is $h = \lg_{3}{n}$
+
+so $T(n) = \text{depth} * cn$ and
+
+$cn \lg_{3/2}{n} \leq T(n) \leq cn \lg_{3}{n} $
+
+therefore $T(n) = \Theta(n\lg n)$
+
+
+