Power law relationships in the branching of three tree species loblolly pine red maple sugar maple
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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 Much diversity follows simple patterns

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Stephen Burton July 24, 2009 AAMU REU

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Power Law Relationships in the Branching of Three Tree Species (Loblolly Pine, Red Maple, Sugar Maple)

Stephen Burton

July 24, 2009


Diversity on Earth

  • Great diversity in organisms and habitats

  • Much diversity follows simple patterns

  • Patterns described by simple mathematical function

  • Known as “power laws”

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

Examples of Power Laws

  • Metabolism and body mass

  • Heart rate, life span, and population growth

  • River systems


Importance of Power Laws

  • Reductionistic science

  • Small scales to large scales

  • Greater understanding of biodiversity

  • Further understanding of underlying principles of math and science

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

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

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


  • 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)

  • Red Maple (Acer rubrum)

  • Sugar Maple (Acer saccharum)

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.

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

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

Sugar Maple

  • Widespread, Great Lakes region

  • 40 meters tall

  • Moist, fertile soil

  • Not adaptable

  • Hard wood for construction and floors

  • Cheap sugar and syrup


  • Healthy and accessible specimens

  • Damaged or too small

  • Too few developed branches

  • Intact canopy

  • Metric tape measure

Measurement Terms

  • L0: trunk of tree

  • L1: first level

  • L2: second level

  • L3: third level

  • L4: fourth level

Data Analysis


Loblolly Pine Results

  • Follow similar power law

  • Very narrow range of values for R-Squared

  • Follow power law very closely

Sugar Maple Results

  • Follow similar power law

  • Very narrow range of values for R-Squared

  • Follow power law very closely

Red Maple Results

  • Weaker power law

  • Very broad range of values for R-Squared

  • Follow power law less closely

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

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


  • 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


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