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We quantify the morphologies and dynamical states of 59 galaxy clusters usingthe power-ratio technique of Buote & Tsai applied to ROSAT PSPC X-ray images.The clusters exhibit a particularly strong $P_2/P_0 - P_4/P_0$ correlation inthe $1h^{-1}_{80}$ Mpc aperture which may be interpreted as an evolutionarytrack; the location of a cluster on the correlation line indicates thedynamical state of the cluster and the distribution of clusters along thistrack measures the rate of formation and evolution of clusters in our sample.The power ratios anti-correlate with the cooling-flow rate indicating areasonable dependence of the flow rate on cluster morphology. The relationshipof the power ratios to the optical Bautz-Morgan (BM) Type is more complex. Thisis because the power ratios are sensitive to unrelaxed regions of clusterswithin a specified scale, whereas BM types are sensitive to unrelaxed regionsover many scales. We discuss further astrophysical applications exploiting therelationship between the power ratios and the evolutionary states of clusters.Comment: 31 pages (8 figures), PostScript, to appear in ApJ on February 10,  1996. In this version we added four more clusters to our sample and revised  the error estimates for the power ratios (and the corresponding figures

Quantifying the Morphologies and Dynamical Evolution of Galaxy Clusters. II. Application to a Sample of ROSAT Clusters