  We address here spanning tree problems on a graph with binary edge weights.
For a general weighted graph the minimum spanning tree is solved in
super-linear running time, even when the edges of the graph are pre-sorted. A
related problem, of finding a spanning tree with a pre-specified sum of
weights, is NP-hard. In contrast, for a graph with binary weights associated
with the edges, it is shown that the minimum spanning tree and finding a
spanning tree with a given total sum, are solvable in linear time with simple
algorithms.
