  We investigate the equilibrium and radial stability of spherically symmetric
relativistic stars, considering a polytropic equation of state (EoS), within
the framework of $f(R,T)$ gravity with a conservative energy-momentum tensor.
Both modified stellar structure equations and Chandrasekhar's pulsation
equations are derived for the $f(R,T)= R+ h(T)$ gravity model, where the
function $h(T)$ assumes a specific form in order to safeguard the conservation
equation for the energy-momentum tensor. The neutron star properties, such as
radius, mass, binding energy and oscillation spectrum are studied in detail.
Our results show that a cusp -- which signals the appearance of instability --
is formed when the binding energy is plotted as a function of the compact star
proper mass. We find that the squared frequency of the fundamental vibration
mode passes through zero at the central-density value corresponding to such a
cusp where the binding energy is a minimum.
