  In this paper, we study the cases when the weighted Bergman metrics of a
domain are invariant under biholomorphisms by introducing the concept of {\it
invariant weight assignments}, focusing on two examples by Tian and Tsuji,
respectively. Using Bergman's minimum integral method and a domain version of
the Tian-Yau-Zelditch expansion for the weighted Bergman kernels and metrics,
we give an alternative proof of uniform convergence of Tian's sequence of
Bergman kernels and metrics on uniform squeezing domains. We also present a
proof of the uniform convergence of Tsuji's dynamical kernel sequence on
uniform squeezing domains.
