Press release from 26 November 2015

Using sphere packing models to explain the structure of forests

New modelling approach to assess the distribution of different tree sizes in natural forests

Explaining the complex structure of tropical forests is one of the great challenges in ecology. An issue of special interest is the distribution of different sizes of trees, something which is of particular relevance for biomass estimates. A team of modellers from the Helmholtz Centre for Environmental Research (UFZ), working together with research partners, has now developed a new method which can be used to explain the tree size distribution in natural forests. To do so, the scientists use principles from stochastic geometry, as they have reported in a contribution to the Proceedings of the National Academy of Sciences of the United States of America (PNAS, Early Edition). Using this approach, it is possible to assess the structure of natural forests across the world more quickly, and produce more accurate biomass estimates.

Image showing tightly packed tree crowns in a natural tropical forest. Photo: André Künzelmann/UFZ

Image showing tightly packed tree crowns in a natural tropical forest, for investigating the forest's structure. Tree crowns of different sizes are shown as spheres.
Photo: André Künzelmann, UFZ


For over one hundred years, the distribution of different sizes of trees in forests has been one of the core attributes recorded by foresters and ecologists world-wide, as it can be used to derive many other structural features, such as biomass and productivity. "We wanted to explain this important pattern", said Dr. Franziska Taubert. Working with her UFZ colleagues Dr. Thorsten Wiegand and Prof. Andreas Huth, and other research partners in the Leipzig University of Applied Sciences (HTWK) and the Karlsruhe Institute of Technology (KIT), they have applied the theory of stochastic sphere packing, which is usually used in physics or chemistry. This theory describes how spheres can be placed in an available space. To apply the theory, the scientists randomly distributed tree crowns of different sizes in forest areas. These tree crowns were not permitted to overlap,  just like packing apples into a box. The distribution of the trees that have been successfully placed in the packing process was then used to determine the tree size distribution.

"Many forest models are based on a dynamic approach: they take into account processes such as growth, mortality, regeneration and competition between trees for light, water and soil nutrients", said Taubert. "These models are complex and data-hungry", added Thorsten Wiegand," so we decided to take a radically different approach, which is fundamentally simpler and only based on spatial structures". This model approach proved its effectiveness by enabling observed forest structures, especially the tree size distribution, to be reproduced accurately. The rules of stochastic geometry are thereby enriched by tree geometry relationships, and the resulting tree packing system is compared to inventory data from tropical forests in Panama and Sri Lanka.

Although one might imagine that a tropical forest is very tightly packed, the scientists came to a surprising conclusion: the packing density of

the tree crowns, which averages 15 to 20%, is astonishingly low. "In particular, the upper and lower canopy levels are less tightly packed with tree crowns", said Taubert. High packing densities of around 60%, which are also possible according to stochastic geometry, only occur at tree heights between 25 and 40 meters.

The findings concerning the distribution of tree crowns are important, because they can be used to draw conclusions about, for example, the carbon content or productivity of a forest. Using this modelling approach, the researchers were also able to show that the decisive factor in shaping the tree size distribution is competition for space. "In classical forest models", said Andreas Huth, "the trees instead compete for light, or water and nutrients".

The theory opens up several new perspectives. The team plans to assess how the model can be applied to natural forests in the temperate and boreal zone. They believe that the model can be used to identify disturbed forests. "That is of special interest because it will enable us to develop a disturbance index", said Taubert, “and to better interpret remote sensing observations by using the structure of natural forests as a reference”. Another benefit of the new theory is that this simple forest packing model takes much less effort than classical forest models. The new approach is an important step toward identifying a minimal set of processes responsible for generating the spatial structure of natural forests.


Franziska Taubert, Markus Wilhelm Jahn, Hans-Jürgen Dobner, Thorsten Wiegand and Andreas Huth:
„The structure of tropical forests and sphere packings“.
Proceedings of the National Academy of Sciences of the United States of America (PNAS).

Involved institutions

  • Helmholtz Centre for Environmental Research – UFZ
  • Leipzig University of Applied Sciences (HTWK)
  • Karlsruhe Institute of Technology (KIT)
  • German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig
  • University of Osnabrück.

The researchers thank the Advanced Grant of the European Research Council (ERC) for their support.

Further information

Dr. Franziska Taubert
Department of Ecological Modelling, UFZ
Phone: +49 341 235-1896

Prof. Dr. Andreas Huth
Deputy of Head of Department of Ecological Modelling, UFZ
Phone: +49 341 235-1719


Susanne Hufe (UFZ-Pressestelle)
Phone: +49 341 235-1630

In the Helmholtz Centre for Environmental Research (UFZ), scientists conduct research into the causes and consequences of far-reaching environmental changes. Their areas of study cover water resources, biodiversity, the consequences of climate change and possible adaptation strategies, environmental technologies and biotechnologies, bioenergy, the effects of chemicals in the environment and the way they influence health, modelling and social-scientific issues. Its guiding principle: Our research contributes to the sustainable use of natural resources and helps to provide long-term protection for these vital assets in the face of global change. The UFZ employs more than 1,100 staff at its sites in Leipzig, Halle and Magdeburg. It is funded by the federal government, Saxony and Saxony-Anhalt.

The Helmholtz Association contributes to solving major and urgent issues in socie-ty, science and industry through scientific excellence in six research areas: Energy, earth and environment, health, key technologies, structure of matter as well as aviation, aerospace and transportation. The Helmholtz Association is the largest scientific organisation in Germany, with 35,000 employees in 18 research centres and an annual budget of around €3.8 billion. Its work is carried out in the tradition of the great natural scientist Hermann von Helmholtz (1821-1894).