  The tropical Grassmannian ${\rm Trop}\, G(2,n)$ is known to be the moduli
space of unrooted metric trees with $n$ leaves. A positive part can be defined
for each of the $(n-1)!/2$ possible planar orderings, $\alpha$, and agrees with
the corresponding planar trees in the moduli space, ${\rm
Trop}^{\alpha}G(2,n)$. Motivated by a physical application we study the way
${\rm Trop}^{\alpha}G(2,n)$ and ${\rm Trop}^{\beta}G(2,n)$ intersect in ${\rm
Trop}\, G(2,n)$. We define their intersection number as the number of unrooted
binary trees that belong to both and construct a $(n-1)!/2\times (n-1)!/2$
intersection matrix. We are interested in finding the diagonal (up to
permutations of rows and columns) submatrices of maximum possible rank for a
given $n$. We prove that such diagonal matrices cannot have rank larger than
$(n-3)!$ using the CHY formalism. We also prove that the bound is saturated for
$n=5$ (the condition is trivial for $n=4$), that for $n=6$ the maximum rank is
$4$, and that for $n=7$ the maximum rank is $\geq 14$. We also ask the
following question: Is there a value $n_{\rm c}$ so that for any $n>n_{\rm c}$
the bound $(n-3)!$ is always saturated? We review and extend two relevant
results in the literature. The first is the Kawai-Lewellen-Tye (KLT) choice of
sets which leads to a $(n-3)!\times (n-3)!$ block diagonal submatrix with
blocks of size $d\times d$ with $d = \lceil (n-3)/2\rceil !\lfloor
(n-3)/2\rfloor !$. The second result is that the number of ${\rm Trop}^\alpha
G(2,n)$'s that intersect a given one grows as $\exp(n\log (3+\sqrt{8}))$ for
large $n$ which implies that the density of the intersection matrix goes as
$\exp(-n(\log(n)-2.76))$. We interpret this as an indication that the generic
behavior is not seen until $n \approx \exp (2.76)$, i.e. $n = 16$. We also find
an exact formula for the number of zeros in a KLT block.
