 Coordinatewise minimum spanning-tree weight vector

Sort the edge weights of a minimum spanning tree $T$ and of any spanning tree $S$ in nondecreasing order. Then $w_i(T)\le w_i(S)$ for every position. To prove it, fix a threshold $t$ and retain the graph's edges of weight at most $t$. If this graph has $c$ connected components, $T$ contains a spanning forest of each component using those edges: otherwise a light edge joining two of its pieces could replace an edge heavier than $t$ on its tree path, decreasing total weight. Hence exactly $|V|-c$ edges of $T$ have weight at most $t$. No other spanning tree can have more, since its retained edges form a forest inside those components. These threshold counts imply the coordinatewise inequality for sorted weights.