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Analysis

T(n) = $\left\{ \begin{array}{ll} \Theta(1) & n=0 \\ T(k) \;+\; T(n-k-1) \;+\; \Theta(1) & n>0 \end{array} \right.$

k = (n-1)/2:

T(n) = T((n-1)/2) + T(n - (n-1)/2 - 1) + $\Theta(1)$

= T((n-1)/2) + T(n/2 + 1/2 - 1) + $\Theta(1)$

= T((n-1)/2) + T((n-1)/2) + $\Theta(1)$

= 2T((n-1)/2) + $\Theta(1)$

$\leq$ 2T(n/2) + $\Theta(1)$

= $\Theta(n)$

k = 0:

T(n) = T(0) + T(n-1) + $\Theta(1)$

= T(n-1) + $\Theta(1)$

= $\Theta(n)$

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