Fig. 1.

i A tree $T\in B(X)$ for $X=\{a,b,c,d,e\}$; ii vertex v is supported by the triple bce (the dashed lines show the edge-disjoint paths from v to these three leaves); iii the cover graph $\mathrm{\Gamma }(\mathcal{T})$ corresponding to the triplet cover $\mathcal{T}$ obtained by taking all pairs from the triple bce that supports v and from the triples abc and cde that support vertices u and w, respectively. This triplet cover is minimal, and since its size is 7 ($=2n-3$ for $n=|X|$) it is also a minimum triplet cover for the tree (by Proposition 3)