The Reiterative Nature of Tree Branches Christine Schneider

Trees and shrubs show a variety of morphologies. There are tall slender trees with predominant main stem with short primary branches.

Other trees have a less dominant main stem with very long branches.

There are many trees with other forms of morphologies.

To date, there has been very little research into a unifying principle of tree and shrub morphologies.

The next few slides show our concept of how branches grow and the series is the basis for our idea that bending stresses of all branches are constant based upon the reiterative process. The red branches are primary branches. The blue branches are secondary branches and green branches are terminal branches.

Hypotheses 1. Mechanical stress is constant from the base to the tip of a branch. 2. All terminal portions of branches will have similar mechanical stress. 3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches. 4. Mechanical stresses of primary branches are constant among tree species.

Mechanical Properties: Bending Moment (M) Wilson and Archer, 1977 The next series of slides gives the definition of bending moment. The next view shows how bending moment is related to size of the branch.

Bending Moment (M) [low] Bending Moment (M) [intermediate] Bending Moment (M) [high]

Mechanical Properties: Section Modulus (S) Beer and Johnston, 1992 The next few slides define section modulus and shows how when a board is vertical the section modulus is different when the board is horizontal.

Mechanical Properties: Stress Beer and Johnston, 1992 Bending stress is the slope of the line of bending moment versus section modulus. Our hypothesis is that bending stress is constant among primary, secondary and terminal branches.

Measurements Diameter of branch (segment) Length of branch (segment) Weight of branch (segment) Weight of side branches

Table 1: Properties of tree branches Species section modulus at tip (m^3 x 10^-8) section modulus at base (m^3 x 10^-8) Acer palmatum Ailanthus altissima Aler carpinfolium Arbutus unedo Betula albosinesis Betula pubescens Broussonetia papyfera Cedrus libani Chranantagus retusus Cornus alternifolia Cornus florida Table 1 shows the section modulii of the primary branches of the study.

Species section modulus at tip (m^3 x 10^-8) section modulus at base (m^3 x 10^-8) Crataegus diffusa Ligustrum lucidum Liquidamber styraciflua Malus hypenhesis Phillyrea latifolia Picea likingensis Picea stchensis Pinus strobus Pinus thunbergii Prunus serotina Quercus palustrius Rhododendron arboreum Tilia tomentosa Ulmus procera

1. Mechanical stress is constant from the base to the tip of a branch. Example 1: Ulmus procera The next two figures show the slope values for primary branches. The slope for Ulmus procera is

1. Mechanical stress is constant from the base to the tip of a branch. Example 2: Pinus thunbergii The slope for Pinus thunbergii is

Speciesr2r2 Linear Equation Acer palmatum0.95y = x Aler carpinfolium0.93y = x Ailanthus altissima0.99y = x Arbutus unedo0.93y = x – Betula albosinesis0.93y = x – Betula pubescens0.85y = 0.105x Broussonetia papyrifera0.93y = x Cedrus libani0.98y = x Chranantagus retusus0.96y = x Cornus alternifolia0.94y = x Cornus florida0.99y = x Crataegus diffus0.97y = x – Ligustrum lucidum0.98y = x – Here we see the equations for primary branches of the species of this study.

Speciesr2r2 Linear Equation Malus hypenhesis0.94y = x Phillyrea latifolia0.99y = x – Picea likingensis0.96y = x Picea stchensis0.87y = x – Pinus strobus0.99y = x Pinus thunbergii0.99y = x Prunus serotina0.95y = 0.194x Quercus palustrius0.98y = x Rhododendron arboreum0.96y = Tilia tomentosa0.97y = x Ulmus procera0.96y = 0.138x Mean0.96 Standard Deviation0.036

2. All terminal portions of branches will have similar mechanical stress. (SSPS Institute Inc, 2000) Species Branch Number Smallest Section Modulus Largest Section Modulus Stress Probability Value Acer palmatum Aler carpinfolium This table over two pages shows that terminal portions of branches had similar bending stresses as expressed by probability values.

Species Branch Number Smallest Section Modulus Largest Section Modulus Probability Value Betula pubescens Crataegus diffusa Ulmus procera

3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches.

SpeciesBranchSM at tipSM at base Stress Probability Acer palmatum A B Acer palmatum A B Ailanthus altissima A B C Aler carpinfolium A B C D Arbutus unedo A B (SSPS Institute Inc, 2000) This table over three pages shows that secondary portions of branches had similar bending stresses as expressed by probability values.

SpeciesBranchSM at tipSM at baseProbability Betula albosinesis A B C Broussonetia papyrifera A B C Cedrus libani A B C Cornus florida A B Liquidamber styraciflua A B C Malus hypnhesis A B

SpeciesBranchSM at tipSM at baseProbability Phillyrea latifolia A B Phillyrea latifolia A B Pinus thunbergii A B C Prunus serotina A B C D Quercus palustrius A B

4. Mechanical stresses of primary branches are constant among tree species. Mean = 5.64, Standard deviation = 1.89 This figure shows that bending stresses of primary branches were relatively constant among the species tested. Other data (not shown) have been procured to demonstrate that bending stresses of primary, secondary, and terminal branches were similar.

Summary 1. Mechanical stress is constant from the base to the tip of a branch. 2. All terminal portions of branches will have similar mechanical stress. 3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches. 4. Mechanical stresses of primary branches are constant among tree species.

Works Cited Beer F.P. and Johnston, E.R. Jr. (1992) Mechanics of materials. 2 nd Edition. McGraw- Hill, Inc. NY Bertram JE, (1989) Size-dependent differential scaling in branches; the mechanical design of trees revisited Trees 4: Castera P, Mortier V. (1991) Growth patterns and bending mechanics of branches Trees 5: Dean T, Roberts S, Gilmore D, Maguire D, Long J, O’Hara K, Seymour R (2002) An evaluation of the uniform stress hypothesis based on stem geometry in select North American conifers. Trees 16: Morgan J, Cannell M. (1987) Structural analysis of tree trunks and branches: tapered cantilever beams subject to large deflections under complex loading. Tree Physiol. 3: Morgan J, Cannell M (1994) Shape of tree stems- a re-examination of the uniform stress hypothesis. Tree Physiol 14, 49 (1994) SPSS Institute Inc Systat 10 [computer program]. SPSS Institute, Chicago, IL. Wilson B, Archer R (1979) Tree design: some biological solutions to mechanical problems. Bioscience 29: Wilson B, Archer R (1977) Reaction wood; induction and mechanical action. Ann. Rev. Plant Physiol. 28:23-43 Zar, J.H. (1974) Biostatistical Analysis. Prentice-Hall, Inc. Englewood Cliffs, NJ