Branching events tend to be clustered around population size changes. For example if Tau is 8, the mode of the mismatch distributions occurs at about 8 differences. It can take large sample sizes to see this. These three observations are what enable Alan Rogers's method of moments to infer population history from the mismatch distribution. Regardless of population history, mutations with low frequencies are more common than mutations with high frequencies.
Population growth causes the appearance of more low-frequency variants.
Population decrease causes the appearance of more high-frequency variants. It can take large sample sizes to see this. With population increase, frequency spectra get more stable.