  Hierarchical Clustering is a popular tool for understanding the hereditary
properties of a data set. Such a clustering is actually a sequence of
clusterings that starts with the trivial clustering in which every data point
forms its own cluster and then successively merges two existing clusters until
all points are in the same cluster. A hierarchical clustering achieves an
approximation factor of $\alpha$ if the costs of each $k$-clustering in the
hierarchy are at most $\alpha$ times the costs of an optimal $k$-clustering. We
study as cost functions the maximum (discrete) radius of any cluster
($k$-center problem) and the maximum diameter of any cluster ($k$-diameter
problem). In general, the optimal clusterings do not form a hierarchy and hence
an approximation factor of $1$ cannot be achieved. We call the smallest
approximation factor that can be achieved for any instance the price of
hierarchy. For the $k$-diameter problem we improve the upper bound on the price
of hierarchy to $3+2\sqrt{2}\approx 5.83$. Moreover we significantly improve
the lower bounds for $k$-center and $k$-diameter, proving a price of hierarchy
of exactly $4$ and $3+2\sqrt{2}$, respectively.
