$($b$)$ Suppose that a forest with $4$ components has $12$ vertices of degree greater than $1,$ and that these degrees add up to $24.$ All of the remaining vertices have degree $1.$ How many vertices of degree $1$ must there be$?$

$($c$)$ Find the long codeword for the spanning tree in $\backslash (\backslash$mathbf$\{$K$\}\_\{6\}\backslash )$ that corresponds to the short codeword AACC and draw a diagram that represents the corresponding spanning tree.

$($d$)$ Use Ford's Algorithm to construct an optimal $\backslash ($ R $\backslash )-$rooted spanning tree in the following weighted directed graph: