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 treeW 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) < FW (d). As a consequence, the use of minimum evolution is consistent provided that the length of each tree W is estimated using a member of U(W ). It is shown how to find many explicit members of U(W ) for each W . The ordinary least-squares expression for W is shown to be in U(W ).