$(3)($i$)$ If a graph with $7$ vertices has degrees $6,4,4,3,2,1,d,$ what is the greatest value that $d$ could have? Draw an example of a graph for which $d$ takes this value.

$($ii$)$ What is the greatest possible number of vertices of degree $1$ in a tree with $11$ vertices such that each vertex has degree at most $4?$ Draw an example of such a graph with this number of vertices of degree $1.$

$($iii$)$ Let $n4$ be an even integer. What is the greatest possible number of edges in a graph with $n$ vertices that has a closed Euler trail? $($Give your answer as an expression in terms of $n.)$ For $n=6,$ draw an example of such a graph with this number of edges.