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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Data Analysis Tree 1 LengthNumber(Number/Total) x 100Log(Length)Log(Number + 1)Log[(Number/Total) x 100] ******************************* ******************************* ******************************* ******************************* *******************************

Graphs

Loblolly Pine Results Follow similar power law Follow similar power law Very narrow range of values for R-Squared Very narrow range of values for R-Squared Follow power law very closely Follow power law very closely Loblolly Pine R-Squared Values Tree Tree Tree Tree Tree

Sugar Maple Results Follow similar power law Follow similar power law Very narrow range of values for R-Squared Very narrow range of values for R-Squared Follow power law very closely Follow power law very closely Sugar Maple R-Squared Values Tree Tree Tree Tree Tree

Red Maple Results Weaker power law Weaker power law Very broad range of values for R-Squared Very broad range of values for R-Squared Follow power law less closely Follow power law less closely Red Maple R-Squared Values Tree Tree Tree Tree Tree Tree Tree

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

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

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

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