In an excerpt of an article by Bernd Sturmfels, I found: 


**Theorem 5.5.** The tropical Grassmannian $G^{′}\_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by $T\_n$ and is defined as follows. The vertex set consists of all unordered pairs $\left \{ A,B \right \}$ where $A$ and $B$ are disjoint subsets of $\left [ n \right ]:=\left \{ 1,2, ... , n \right \}$ having cardinality at least two, and $A \cup B=\left [ n \right ]$. Such pairs are called splits. The number of splits is $2^{n−1} − n − 1$. Two splits $\left \{ A,B \right \}$ and $\left \{ A^{'},B^{'} \right \}$ are connected by an edge in the simplicial complex $T\_n$ if and only if


(5) $A \sqsubseteq {A}'$ or $A \sqsubseteq {B}'$ or $B \sqsubseteq {A}'$ or $B \sqsubseteq {B}'$. 


We define $T\_n$ as the largest simplicial complex having this edge graph.... In the language of algebraic combinatorics, $T\_n$ is the flag complex of the compatibility graph specified by (5) on the set of all $2^{n−1} − n − 1$ splits. 


**Example 5.6.** ($n = 6$) The two-dimensional simplicial complex $T\_6$ has $25$ vertices, $105$ edges and $105$ triangles... 


Question: Are $56, 490, 1260, 945$ the "face" numbers for $T\_7$?



"Face" numbers for tropical Grassmannian G′_2,7 simplical complex ?
You have the right numbers. $\mathrm{Trop} \ G(2,n)$ is the space of phylogenetic trees studied in [Billera-Holmes-Vogtman.](http://www-stat.stanford.edu/~susan/papers/lap.pdf) It has $1 \times 3 \times 5 \times \cdots \times (2n-5)$ maximal faces and $2^{n-1} - n-1$ vertices, matching your $945$ and $56$. 


The first several face numbers are
$$\begin{matrix}
1 & & & & \\
1 & 3 & & & \\
1 & 10 & 15 & & \\
1 & 25 & 105 & 105 & \\
1 & 56 & 490 & 1260 & 945 \\
\end{matrix}$$


Write $f(n,k)$ for the number of $k$ faces in $\mathrm{Trop} \ G(2,n)$, we have
$$f(n,k) = (k+1) f(n-1,k) + (n+k-2) f(n-1,k-1).$$
I am prone to off by one errors, so check this against the above table.


It is easy to get closed formulas for $f(n,r)$ or $f(n,n-r)$, for small fixed $r$, but there seems to be no simple expression for the middle terms. Billera-Holmes-Vogtman say that there is more discussion in Exercise 5.40 of *Enumerative Combinatorics I* by Stanley; I don't have my copy with me to check.


