Abstract: Forest size structure (the diameter distribution of trees in a forest) is a comprehensive reflection of forest demographic processes. It is the basis for determining forest successional stage and the state of forest health, estimating forest biomass and predicting forest carbon sink potential. Studies of forest size structure began with statistical descriptions and then progressed to theoretical and mathematical deductions. In early statistical studies of forestry, many common probability distribution functions were used for fitting the plot scale size structure variation, yet most of these functions were not derived from biological processes and therefore lack clear biological meanings. With the development of macroecology, the principle of maximum entropy and the central limit theorem have been used to explain forest size structure at large spatial scales which is found to be relatively consistent. Such models mainly focus on probability statistics rather than ecological processes. In 2000s, the finding of power-law size structure in natural mature forests has spawned a series of theoretical studies, including the metabolic scaling theory, the theory of gap succession, etc. These theories have tried to attribute the formation of power-law size structure to the relationship between tree size and resource use on individual scale and tree competition for resources on community scale. Demogaphic equilibrium theory provides a general framework for analyzing the relationship between the steady state forest size structure and tree growth and mortality. Under this equilibrium framework, the hypothesis of demographic optimality further provides a new perspective for the analysis of forest size structure. Mathematical models including transition matrices, integral projections, and partial differential equations are powerful tools for the analysis of forest size structure dynamics. However, due to the difficulty in giving time-dynamic solutions to the mathematical models, most studies have been confined within the framework of forest demographic equilibrium. In order to understand the dynamic patterns of forest size structure and predict forest carbon sink potential in a rapidly changing climate, it is essential to find a general time-dynamic solution to the mathematical models under non-steady-state conditions and determine the effects of climatic factors on forest growth and mortality rates.

Key words: forest size structure, resource competition, tree growth and mortality, carbon sink potential