  We define a new balance index for rooted phylogenetic trees based on the
symmetry of the evolutive history of every set of 4 leaves. This index makes
sense for multifurcating trees and it can be computed in time linear in the
number of leaves. We determine its maximum and minimum values for arbitrary and
bifurcating trees, and we provide exact formulas for its expected value and
variance on bifurcating trees under Ford's $\alpha$-model and Aldous'
$\beta$-model and on arbitrary trees under the $\alpha$-$\gamma$-model.
