$-[145][246][356][456], [146][156][246][356], [145][245][256][356],$

$[145][246][346][345], [126]^2[156][356], [125]^2[256][356], [134]^2[246][346],$

$[136]^2[146][246], [145][245][235]^2, [145][345][234]^2,$

$[136]^2[126]^2, [125]^2[235]^2, [134]^2[234]^2, [123]^4$

where we observe how a zero area, like $[135]$, occurs as an unseen $1$ due to the zero volume exponent. These vertices thus depend on the metric structure of the triangle with vertices at $1$, $2$ and $3$. The so called toric specialisation takes bracket terms to monomials in $c_i$, the six coefficients in a quadratic form $f(X,Y,Z)$, via

$[i_1 i_2 \cdots i_k] \mapsto \textrm{det}(i_1, i_2, \cdots, i_k) c_1 c_2 \cdots c_k$

Here we see the volume factor go to an entropic coefficient, as in the term $[123]^4$ giving $256 c_{1}^{4}c_{2}^{4}c_{3}^{4}$. The sum of all $28$ such determinant terms turns out to be the product of principal minors for a $3 \times 3$ symmetric matrix with diagonal $2c_1, 2c_2, 2c_3$ and off diagonal terms $c_4$, $c_5$ and $c_6$. Thus a $3 \times 3$ matrix is associated to the six degree $2$ three qutrit functions.

Observe how the bracket monomials above could

*actually resemble*associations of five objects, which usually label the associahedron by trees dual to the triangulated polygon. There are four nodes on a five leaved tree, and a total exponent of four on the bracket monomials. A tree node is labeled by a bracket, but a given bracket is allowed to repeat itself. For instance, if we imagine that $[123]$ labels the bottom node of the special tree for $[123]^4$, then $[123]$ stands for the fact that all subtrees above and to the left of the node are full binary trees. Then the mirror tree would be given by the term $-[145][246][356][456]$, and the spreading out of the $1,2,3$ reflects the fact that the left branch is now a minimal tree. Here we see

*metric structure*being associated to tree shapes.

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