  Zipf's law of city-size distributions can be expressed by three types of
mathematical models: one-parameter form, two-parameter form, and
three-parameter form. The one-parameter and one of the two-parameter models are
familiar to urban scientists. However, the three-parameter model and another
type of two-parameter model have not attracted attention. This paper is devoted
to exploring the conditions and scopes of application of this Zipf models. By
mathematical reasoning and empirical analysis, new discoveries are made as
follows. First, if the size distribution of cities in a geographical region
cannot be described with the one- or two-parameter model, maybe it can be
characterized by the three-parameter model with a scaling factor and a
scale-translational factor. Second, all these Zipf models can be unified by
hierarchical scaling laws based on cascade structure. Third, the patterns of
city-size distributions seems to evolve from three-parameter mode to
two-parameter mode, and then to one-parameter mode. Four-year census data of
China's cities are employed to verify the three-parameter Zipf's law and the
corresponding hierarchical structure of rank-size distributions. This study is
revealing for people to understand the scientific laws of social systems and
the property of urban development.
