  $\newcommand{\Max}{\mathrm{Max4PC}}$ The Four point condition (4PC
henceforth) is a well known condition characterising distances in trees $T$.
Let $w,x,y,z$ be four vertices in $T$ and let $d_{x,y}$ denote the distance
between vertices $x,y$ in $T$. The 4PC condition says that among the three
terms $d_{w,x} + d_{y,z}$, $d_{w,y} + d_{x,z}$ and $d_{w,z} + d_{x,y}$ the
maximum value equals the second maximum value.
  We define an $\binom{n}{2} \times \binom{n}{2}$ sized matrix $\Max_T$ from a
tree $T$ where the rows and columns are indexed by size-2 subsets. The entry of
$\Max_T$ corresponding to the row indexed by $\{w,x\}$ and column $\{y,z\}$ is
the maximum value among the three terms $d_{w,x} + d_{y,z}$, $d_{w,y} +
d_{x,z}$ and $d_{w,z} + d_{x,y}$. In this work, we determine basic properties
of this matrix like rank, give an algorithm that outputs a family of bases, and
find the determinant of $\Max_T$ when restricted to our basis. We further
determine the inertia and the Smith Normal Form (SNF) of $\Max_T$.
