  Using the path integral approach, we obtain the characteristic functions of
the gyration radius distributions for Gaussian star and Gaussian rosette
macromolecules. We derive the analytical expressions for cumulants of the both
distributions. Applying the steepest descent method, we estimate the
probability distribution functions of the gyration radius in the limit of a
large number of star and rosette arms in two limiting regimes: for strongly
expanded and strongly collapsed macromolecules. We show that in both cases, in
the regime of a large gyration radius relative to its mean-square value, the
probability distribution functions can be described by the Gaussian functions.
In the shrunk macromolecule regime, both distribution functions tend to zero
faster than any power of the gyration radius. Based on the asymptotic behavior
of the distribution functions and the behavior of statistical dispersions, we
demonstrate that the probability distribution function for the rosette is more
densely localized near its maximum than that for the star polymer. We construct
the interpolation formula for the gyration radius distribution for the Gaussian
star macromolecule which can help to take into account the conformational
entropy of the flexible star macromolecules within the Flory-type mean-field
theories
