Let T be a $(2$k $+1)-$node BST with nodes x x$2$k$+1,$and where xi is the left child of node xi$+1$ for all i in $\{1,...,2$k$\}$ Let T $($resp$,$ T $)$ be the BST obtained from T by rotating $($resp$.,$ splaying$)$ x$1$ to the root.

$($a$)(4$ points$)$ Determine the structure of T $$ and T $.$

$($b$)(6$ points$)$ Assuming that we assign unit weight to each node, determine a $\backslash$Theta $-$bound on the decrease in potential from T to T $.$