  High density clusters can be characterized by the connected components of a
level set $L(\lambda) = \{x:\ p(x)>\lambda\}$ of the underlying probability
density function $p$ generating the data, at some appropriate level
$\lambda\geq 0$. The complete hierarchical clustering can be characterized by a
cluster tree ${\cal T}= \bigcup_{\lambda} L(\lambda)$. In this paper, we study
the behavior of a density level set estimate $\widehat L(\lambda)$ and cluster
tree estimate $\widehat{\cal{T}}$ based on a kernel density estimator with
kernel bandwidth $h$. We define two notions of instability to measure the
variability of $\widehat L(\lambda)$ and $\widehat{\cal{T}}$ as a function of
$h$, and investigate the theoretical properties of these instability measures.
