Question

(15 pts) 1. Create the recursion tree for the recurrence T(n)-T(2n/5)T3n/5) O(n). Show total complexity

Answer 1

Recursion trees can be useful for gaining intuition about the closed form of a recurrence, but they are not a proof

Recursion Tree for the recurrence

$T(n)=T(\frac{2n}{5})+T(\frac{3n}{5})+O(n)$

can be shown as,

Starting the root as, n, then expnading out to the first level be

Similarly expanding the above tree for further more levels gives,

Note that the tree here is not balanced: the longest path is the rightmost one, and its length is log_5/3 n.

Hence our guess for the closed form of complexity of this recurrence is O(n log n).