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The tropical Grassmannian ${\rm Trop}\, G(2,n)$ is known to be the modulispace of unrooted metric trees with $n$ leaves. A positive part can be definedfor each of the $(n-1)!/2$ possible planar orderings, $\alpha$, and agrees withthe corresponding planar trees in the moduli space, ${\rmTrop}^{\alpha}G(2,n)$. Motivated by a physical application we study the way${\rm Trop}^{\alpha}G(2,n)$ and ${\rm Trop}^{\beta}G(2,n)$ intersect in ${\rmTrop}\, G(2,n)$. We define their intersection number as the number of unrootedbinary 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 topermutations of rows and columns) submatrices of maximum possible rank for agiven $n$. 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 thefollowing 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 relevantresults in the literature. The first is the Kawai-Lewellen-Tye (KLT) choice ofsets which leads to a $(n-3)!\times (n-3)!$ block diagonal submatrix withblocks 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 ${\rm Trop}^\alphaG(2,n)$'s that intersect a given one grows as $\exp(n\log (3+\sqrt{8}))$ forlarge $n$ 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 genericbehavior is not seen until $n \approx \exp (2.76)$, i.e. $n = 16$. We also findan exact formula for the number of zeros in a KLT block.

Diagonally Embedded Sets of ${\rm Trop}^+G(2,n)$'s in ${\rm Trop}\, G(2,n)$: Is There a Critical Value of $n$?