Suppose $\kappa$ is a supercompact cardinal. It is known that for every $\lambda \geq \kappa$, many normal ultrafilters on $P_\kappa(\lambda)$ have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree $T$ of normal ultrafilters associated with $\kappa$. We investigate the distribution throughout $T$ of normal ultrafilters with and normal ultrafilters without the partition property.