  Hierarchies can be modeled by a set of exponential functions, from which we
can derive a set of power laws indicative of scaling. The solution to a scaling
relation equation is always a power law. The scaling laws are followed by many
natural and social phenomena such as cities, earthquakes, and rivers. This
paper is devoted to revealing the power law behaviors in systems of natural
cities by reconstructing the hierarchy with cascade structure. The cities of
America, Britain, France, and Germany are taken as examples to make empirical
analyses. The hierarchical scaling relations can be well fitted to the data
points within the scaling ranges of the size and area of the natural cities.
The size-number and area-number scaling exponents are close to 1, and the
allometric scaling exponent is slightly less than 1. The results show that
natural cities follow hierarchical scaling laws and hierarchical conservation
law very well. The hierarchical scaling law proved to be derived from entropy
maximization principle, and this suggests that the evolution of natural cities
is dominated by entropy maximization laws. This study is helpful for scientists
to understand the power law behavior in the development of cities and systems
of cities.
