Let $\mathbf{A}_{n-1}^+ \subset \mathbb{Z}^n$ denote the set of positive roots
of the root system $\mathbf{A}_{n-1}$ and $I_{\mathbf{A}_{n-1}^+}$
its toric ideal.
The purpose of the present paper is to study combinatorics and algebra
on $\mathbf{A}_{n-1}^+$ and $I_{\mathbf{A}_{n-1}^+}$.
First, it will be proved that $I_{\mathbf{A}_{n-1}^+}$
induces an initial ideal $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$ which is
generated by quadratic squarefree monomials together with
cubic squarefree monomials.
Second, we will associate each maximal face $\sigma$ of
the unimodular triangulation $\Delta$ arising
from $\mathit{in}_{<}\left(I_{\mathbf{A}_{n-1}^+}\right)$
with a certain subgraph $G_\sigma$ on $[n] = \{1,\ldots,n\}$.
Third, noting that the number of maximal faces of $\Delta$
is equal to that of anti-standard trees $T$ on $[n]$
with $T \neq \{ (1,2) , (1,3), \ldots , (1,n) \}$,
an explicit bijection between the set
$\{ G_\sigma \colon \sigma\ \text{is a maximal face of}\ \Delta \}$
and that of anti-standard trees $T$ on $[n]$
with $T \neq \{ (1,2), (1,3), \ldots , (1,n) \}$ will be constructed.
In particular,
a new combinatorial expression of Catalan numbers arises.