S$($n$)=3S(n-1)+2,S(1)=2$

$($a$)$ Draw a recursion tree for S$($n$).$ Include the recursive terms for the nodes in a few levels, and also show what the total non$-$recursive term amounts to for each of those levels. State what the depth of the tree is$,$ and explain how you determined this.

Provide a convincing argument that S$($n$)=\backslash$Theta $($g$($n$)),$ for some simple function g$($n$).$

$($b$)$ Use induction to prove that S$($n$)=$O$($g$($n$)),$ for the same g$($n$)$ as in part $($a$).$