  We obtain a nigh optimal estimate for the first eigenvalue of two natural
weighted problems associated to the bilaplacian (and of a continuous family of
fourth-order elliptic operators in dimension $2$) in degenerating annuli (that
are central objects in bubble tree analysis) in all dimension. The estimate
depends only on the conformal class of the annulus. We also show that in
dimension $2$ and dimension $4$, the first eigenfunction (of the first problem)
is never radial provided that the conformal class of the annulus is large
enough. The other result is a weighted Poincar\'e-type inequality in annuli for
those fourth-order operators. Applications to Morse theory are given.
