  A comparative study of one-dimensional quantum structures which allow
analytic expressions for the position and momentum R\'{e}nyi $R(\alpha)$ and
Tsallis $T(\alpha)$ entropies, focuses on extracting the most characteristic
physical features of these one-parameter functionals. Consideration of the
harmonic oscillator reconfirms a special status of the Gaussian distribution:
at any parameter $\alpha$ it converts into the equality both R\'{e}nyi and
Tsallis uncertainty relations removing for the latter an additional requirement
$1/2\leq\alpha\leq1$ that is a necessary condition for all other geometries. It
is shown that the lowest limit of the semi infinite range of the dimensionless
parameter $\alpha$ where \emph{momentum} components exist strongly depends on
the \emph{position} potential and/or boundary condition for the \emph{position}
wave function. Asymptotic limits reveal that in either space the entropies
$R(\alpha)$ and $T(\alpha)$ approach their Shannon counterpart, $\alpha=1$,
along different paths. Similarities and differences between the two entropies
and their uncertainty relations are exemplified. Some unsolved problems are
pointed at too.
