The sum of distances between vertices of a tree has been considered from many
aspects. The question of characterizing the extremal trees that maximize or minimize
various such “distance-based” graph invariants has been extensively studied. Such
invariants include, to name a few, the sum of distances between all pairs of
vertices and the sum of distances between all pairs of leaves. With respect to
the distances between internal vertices, we provide analogous results that
characterize the extremal trees that minimize the value of any nonnegative
and nondecreasing function of internal distances among trees with various
constraints.