Power Law Relationships in Tree Branching of Loblolly Pine, Red Maple, Sugar Maple

1. Power Law Relationships in the Branching of Three Tree Species (Loblolly Pine, Red Maple, Sugar Maple) Stephen Burton July 24, 2009 AAMU REU

2. Diversity on Earth • Great diversity in organisms and habitats • Much diversity follows simple patterns • Patterns described by simple mathematical function • Known as “power laws”

3. What Are Power Laws? • Ecological patterns that repeat themselves over broad scales • “General features of complex systems” • Limits of power laws from natural and mathematical laws • Patterns from power laws visible in organization of natural systems

4. Examples of Power Laws • Metabolism and body mass • Heart rate, life span, and population growth • River systems • TREE STRUCTURE

5. Importance of Power Laws • Reductionistic science • Small scales to large scales • Greater understanding of biodiversity • Further understanding of underlying principles of math and science

6. Power Laws in Trees • Trunk diameter and branch diameter • Coniferous trees • Density of branches and unknown variable • Independent of environment so internal factor • Suggests a power law is present

7. Tree Branches • Organization of leaves and support • Interdependent networks that maximize the health of the whole tree • Growth patterns reflect best interest of entire tree • Result of evolutionary pressures or functional requirements • Branches are network, so power law

8. Power Laws in Branches • Shoot growth in pine species • Hierarchal growth: low growth in higher order branches • Internal control maintains pattern • Likely a result of a power law

9. Objective • The primary goal of this study was to provide evidence that the branching patterns of trees follow a power law

10. Species Studied • Loblolly Pine (Pinus taeda) • Red Maple (Acer rubrum) • Sugar Maple (Acer saccharum)

11. Loblolly Pine • Southeastern U.S., Texas, Delaware • Second largest range • Coniferous • 30 meters tall • Continuous growth in diameter • Most widely used timber species in U.S.

12. Ecological Importance • Maintain balance of ecosystems • Adaptable, grows and reproduces quickly • Restore areas harmed by fire or logging • Site restoration due to litter layer • Animal habitats

13. Red Maple • Eastern U.S. to New Mexico and Dakotas • Deciduous • 30 to 40 meters tall • Acidic soil and marshes, but adaptable • Utensils and tools • Tourism

14. Sugar Maple • Widespread, Great Lakes region • 40 meters tall • Moist, fertile soil • Not adaptable • Hard wood for construction and floors • Cheap sugar and syrup

15. Measurements • Healthy and accessible specimens • Damaged or too small • Too few developed branches • Intact canopy • Metric tape measure

16. Measurement Terms • L0: trunk of tree • L1: first level • L2: second level • L3: third level • L4: fourth level

17. Data Analysis

18. Graphs

19. Loblolly Pine Results • Follow similar power law • Very narrow range of values for R-Squared • Follow power law very closely

20. Sugar Maple Results • Follow similar power law • Very narrow range of values for R-Squared • Follow power law very closely

21. Red Maple Results • Weaker power law • Very broad range of values for R-Squared • Follow power law less closely

22. Study Expectations • All three species would follow power law • Tree branches are a network • Networks often governed by power laws • Other experiments have confirmed power laws in other species • Two of three species followed this pattern

23. Red Maple Explanation • Did not follow power law closely, but was expected to • Red maples in study were not fully mature • Approximately 2 to 3 years old • Canopy heights were similar to height of surrounding community • Still strongly competing with surrounding species for resources

24. Conclusion • Two of most common species in U.S. follow power law closely • Immature trees do not follow power laws as strongly as mature trees • For power laws, small scale research allows predictions for large scale patterns • Predictive power of science increases

25. References • Adam, J. A. (2003). Mathematics in Nature: Modeling Patterns in the Natural World. Princeton: Princeton Press. • Brown, J. H., V. K. Gupta, B. Li, B. T. Milne, C. Restrepo, and G. B. West. 2002. The fractal nature of nature: power laws, ecological complexity and biodiversity. The Royal Society. 357: 619-626. • Chen, X. and B. Li. 2003. Testing the allometric scaling relationships with seedlings of two tree species. ACTA OECOLOGICA. 24: 125-129. • Coomes, D. A., K. L. Jenkins, and L. E. S. Cole. 2007. Scaling of tree vascular transport systems along gradients of nutrient supply and altitude. biology letters. 3: 86-89. • Gelderen, D. M. van. (1994). Maples of the world. Portland: Timber Press. • Kaiteniemi, P. and A. Lintunen. 2008. Precision of allometric scaling equations for trees can be improved by including the effect of ecological interactions. Trees. 22: 579- 584. • Le Hardy de Beaulieu, Antoine. (2003). An illustrated guide to maples. Portland: Timber Press. • Schultz, R. P. (1997). Loblolly Pine: The Ecology and Culture of Loblolly Pine (Pinus taeda L.). Washington, D. C.: U. S. Department of Agrciculture, Forest Service. • Suzuki, A. A. and M. Suzuki. 2009. Why do lower order branches show greater shoot growth than higher order branches? Considering space availability as a factor affecting shoot growth. Trees. 23: 69-77. • Watt, M. S., J. R. Moore, and B. McKinlay. 2005. The influence of wind on branch characteristics of Pinus radiata. Trees. 19:58-65.