  We model the formation and evolution of galaxy clusters in the framework of
an extended dark matter halo merger-tree algorithm that includes baryons and
incorporates basic physical considerations. Our modified treatment is employed
to calculate the probability density functions of the halo concentration
parameter, intracluster gas temperature, and the integrated Comptonization
parameter for different cluster masses and observation redshifts. Scaling
relations between cluster mass and these observables are deduced that are
somewhat different than previous results. Modeling uncertainties in the
predicted probability density functions are estimated. Our treatment and the
insight gained from the results presented in this paper can simplify the
comparison of theoretical predictions with results from ongoing and future
cluster surveys.
