  We consider ground states of rotating Bose-Einstein condensates with
attractive interactions in non-radially harmonic traps $V(x)=x_1^2+\Lambda
^2x_2^2 $, where $0<\Lambda \not =1$ and $x=(x_1, x_2)\in R^2$. For any fixed
rotational velocity $0\le \Omega <\Omega ^*:=2\min \{1, \Lambda\}$, it is known
that ground states exist if and only if $ a<a^*$ for some critical constant
$0<a^*<\infty$, where $a>0$ denotes the product for the number of particles
times the absolute value of the scattering length. We analyze the asymptotic
expansions of ground states as $a\nearrow a^*$, which display the visible
effect of $\Omega$ on ground states. As a byproduct, we further prove that
ground states do not have any vortex in the region $R(a):=\{x\in R^2:\,|x|\le C
(a^*-a)^{-\frac{1}{12}}\}$ as $a\nearrow a^*$ for some constant $C>0$, which is
independent of $0<a<a^*$.
