Consistent formulas for estimating the total lengths of trees

Reconstructing phylogenetic trees using the criterion of minimum evolution requires the use of a formula FT(d) that estimates the total length of a tree T given only the estimated distances d between the leaves of the tree. Let U(T) be the collection of linear formulas FT(d) that correctly estimate the total length of T whenever d is an additive distance function on T. The current paper characterizes a subset U+(W) of U(W) for each tree W such that whenever T is a completely resolved tree, FT is in U(T), FW is in U+(W), and d is positively additive on T, then FT(d)