  We adopted and extended an efficient Gr\"uneisen formalism to study the
phonon anharmonicity and linear thermal expansion coefficients (TECs) of
trigonal bismuth selenide (Bi$_2$Se$_3$) and antimony telluride (Sb$_2$Te$_3$).
Anharmonicity of the systems is studied via extensive calculation of
Gr\"uneisen parameters that exploit symmetry-preserving deformations.
Consistent with experimental findings, a large anisotropy between the TECs in
the $a$ and $c$ directions is found. The larger anharmonicity inherent in
Sb$_2$Te$_3$, as compared to Bi$_2$Se$_3$ is offset by the volumetric effect,
resulting in comparable temperature dependence of their linear TECs. The Debye
temperatures deduced from our first-principles data also agree very well with
the existing tabulated values. The highly efficient methodology developed in
this work, applied for the first time to study the linear TECs of two trigonal
thermoelectric systems, opens up exciting opportunities to address the
anharmonic effects in other thermoelectrics and other low-symmetry materials.
