  For a reduced word ${\bf i}$ of the longest element in the Weyl group of
$\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$
which parametrizes the dual canonical bases. In this paper, we classify all
${\bf i}$'s such that $C_{\bf i}$ is simplicial. We also prove that for any
regular dominant weight $\lambda$ of $\mathfrak{sl}_{n+1}(\mathbb{C})$, the
corresponding string polytope $\Delta_{\bf i}(\lambda)$ is unimodularly
equivalent to the Gelfand-Cetlin polytope associated to $\lambda$ if and only
if $C_{\bf i}$ is simplicial. Thus we completely characterize Gelfand-Cetlin
type string polytopes in terms of ${\bf i}$.
