Optimality conditions for Hunter’s bound

The bound known as Hunter’s bound states that P(A1∪⋯∪An)≤∑i=1npi−∑{i,j}∈Tpi,j, where TT designates the heaviest spanning tree of the graph on nn nodes with edge weights pi,jpi,j. We prove that Hunter’s bound is optimal if and only if the input probabilities are given on a tree.