  Finite metric trees are known to have strict 1-negative type. In this paper
we introduce a new family of inequalities that quantify the extent of the
"strictness" of the 1-negative type inequalities for finite metric trees. These
inequalities of "enhanced 1-negative type" are sufficiently strong to imply
that any given finite metric tree must have strict p-negative type for all
values of p in an open interval that contains the number 1. Moreover, these
open intervals can be characterized purely in terms of the unordered
distribution of edge weights that determine the path metric on the particular
tree, and are therefore largely independent of the tree's internal geometry.
  From these calculations we are able to extract a new non linear technique for
improving lower bounds on the maximal p-negative type of certain finite metric
spaces. Some pathological examples are also considered in order to stress
certain technical points.
