Suppose we give distinct weights to each of the edges of a tree on at least $2$ vertices. Of course, the minimum spanning tree consists of all of the edges, but Boruvka$$s algorithm may take a varying number of iterations to find it$.$

$($a$)$ Give an edge labeling of the path graph on $8$ vertices where Boruvka$$s algorithm terminates after exactly $3$ iterations

b$)$ For each n$>=2,$ give an example of a tree on n vertices where Boruvka$$s algorithm always terminates in one iteration, regardless of the edge weights.