$(10$ points$)$ Recall the Tree Growing Procedure $($TGP$)$: We start with a single node. Then, zero or more times, we a new node and connect with an existing node by an edge.

$($a$)(5$ points$)$ Show that every graph obtained by the TGP is a tree.

$($b$)(5$ points$)$ Show that for any tree with $n$ nodes and $e$ edges, $e=n-1.$ You may use without proof the fact that every tree can be constructed with the TGP$.$