Let P(c) = P(X
1 ≦ c
1, · ··, Xp
≦ cp
) for a random vector (X
1, · ··, Xp
). Bounds are considered of the form
where T is a spanning tree corresponding to the bivariate probability structure and di
is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that
alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.