A tropical interpretation of m-dissimilarity maps

Let T be a weighted tree with n   numbered leaves and let D=(D(i,j))i,jD=(D(i,j))i,j be its distance matrix, so D(i,j)D(i,j) is the distance between the leaves i and j. If m   is an integer satisfying 2⩽m⩽n2⩽m⩽n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D  . For m=3m=3, we present a tropical description of the set of m  -dissimilarity maps of trees. For m=4m=4, a partial result is given.