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The search for simple principles underlying the complex architecture of ecological communities such as forests still challenges ecological theorists. We use tree diameter distributions-fundamental for deriving other forest attributes-to describe the structure of tropical forests. Here we argue that tree diameter distributions of natural tropical forests can be explained by stochastic packing of tree crowns representing a forest crown packing system: a method usually used in physics or chemistry. We demonstrate that tree diameter distributions emerge accurately from a surprisingly simple set of principles that include site-specific tree allometries, random placement of trees, competition for space, and mortality. The simple static model also successfully predicted the canopy structure, revealing that most trees in our two studied forests grow up to 30-50 m in height and that the highest packing density of about 60% is reached between the 25-and 40-m height layer. Our approach is an important step toward identifying a minimal set of processes responsible for generating the spatial structure of tropical forests.tropical forest | forest size structure | stochastic geometry | tree crown packing | leaf area F orests are one of the world's best investigated ecosystems (1-4). However, despite all of the studies devoted to forests, mechanistic connections among important features of forest physiognomy are not fully understood. Of crucial importance for this are tree diameter distributions, which have been used for decades in ecology and forestry to characterize the state of forests. Tree diameter distributions are available for many forests of the world and allow together with tree allometries prediction of other important forest attributes like leaf area (5), basal area (6), above-ground biomass (7), tree density, and the presence or absence of disturbances (8). Tropical forests usually include many small trees and far fewer large ones (1, 9). Diameter distributions can also be predicted from dynamic forest models (10-13) in which the forest structure emerges from the interplay between the dynamic processes of mortality, regeneration, competition, and growth.In this study, we argue that tree diameter distributions of natural tropical forests can be predicted by stochastic packing theory-a method usually used in physics or chemistry-together with sitespecific tree allometries. Packing systems have a long history of use across a wide range of disciplines, going back as far as Kepler's analysis of regular packing of spheres (i.e., with a maximum of 74% filled volume). Irregular stochastic packing of spheres has been analyzed in the last decades and shows lower maximal packing densities (e.g., jammed sphere packing systems in physics, with 55-64% filled volume) (14, 15). We show here that simple principles of stochastic packing theory together with tree allometries and other simple structural rules allow for an accurate prediction of observed tree diameter distributions and related structural attributes for two tropical for...

Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation – the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.

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