DFS & BFS Spanning Trees

Question

I want to construct a DFS and a BFS spanning trees for the graph below. The root is node a. At each step the next edge to be traversed should be the cheapest one.

DFS:

My understanding that to the construct this DFS where each next is the cheapest one, the resultant spanning tree would be:

{(a,c), (c,h), (h,g), (g,f), (f,b), (b,k), (k, j), (j,i), (i, l), (l,m), (m,e), (e,d)}

BFS:

My understanding that to the construct this BFS where each next is the cheapest one, the resultant spanning tree would be:

Queue: a c d b e f h g k m i j k

Is my understanding of spanning trees correct when the next edge to be traversed should be the cheapest one? Is this a MCST?