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The Reiterative Nature of Tree Branches Christine Schneider

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

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

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There are many trees with other forms of morphologies.

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

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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.

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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.

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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.

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

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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.

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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.

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

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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 palmatum0.47236.8 Ailanthus altissima12.689.7 Aler carpinfolium0.04257.9 Arbutus unedo1.08746.5 Betula albosinesis0.30450.6 Betula pubescens0.11920.3 Broussonetia papyfera0.84897.4 Cedrus libani0.85477.5 Chranantagus retusus3.9641.7 Cornus alternifolia1.027.2 Cornus florida0.38216 Table 1 shows the section modulii of the primary branches of the study.

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Species section modulus at tip (m^3 x 10^-8) section modulus at base (m^3 x 10^-8) Crataegus diffusa0.1814.2 Ligustrum lucidum0.536.3 Liquidamber styraciflua0.86121 Malus hypenhesis8.2185 Phillyrea latifolia2.186.7 Picea likingensis1.116.4 Picea stchensis0.135.7 Pinus strobus0.3316 Pinus thunbergii0.6670.8 Prunus serotina0.4227.4 Quercus palustrius1.8108 Rhododendron arboreum0.9417.1 Tilia tomentosa2.6726 Ulmus procera0.3427.4

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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 0.138.

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1. Mechanical stress is constant from the base to the tip of a branch. Example 2: Pinus thunbergii The slope for Pinus thunbergii is 0.0246.

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Speciesr2r2 Linear Equation Acer palmatum0.95y = 0.0551x + 0.689 Aler carpinfolium0.93y = 0.09.48x + 0.145 Ailanthus altissima0.99y = 0.0875x - 2.40 Arbutus unedo0.93y = 0.0227x – 0.366 Betula albosinesis0.93y = 0.0672x – 0.118 Betula pubescens0.85y = 0.105x - 0.827 Broussonetia papyrifera0.93y = 0.0555x - 1.68 Cedrus libani0.98y = 0.0338x + 1.05 Chranantagus retusus0.96y = 0.0335x + 1.63 Cornus alternifolia0.94y = 0.0531x + 1.3321 Cornus florida0.99y = 0.0709x + 0.8411 Crataegus diffus0.97y = 0.0447x – 0.0523 Ligustrum lucidum0.98y = 0.0604x – 0.0774 Here we see the equations for primary branches of the species of this study.

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Speciesr2r2 Linear Equation Malus hypenhesis0.94y = 0.0307x + 6.2945 Phillyrea latifolia0.99y = 0.0609x – 0.824 Picea likingensis0.96y = 0.0857x - 0.7211 Picea stchensis0.87y = 0.0475x – 0.0881 Pinus strobus0.99y = 0.0917x - 1.39 Pinus thunbergii0.99y = 0.0246x + 0.122 Prunus serotina0.95y = 0.194x - 3.80 Quercus palustrius0.98y = 0.0441x + 0.212 Rhododendron arboreum0.96y = 0.0985 - 0.878 Tilia tomentosa0.97y = 0.0388x + 0.475 Ulmus procera0.96y = 0.138x - 3.41 Mean0.96 Standard Deviation0.036

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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 11.9918.9 20.4720.10.12 32.0212.9 40.9310.6 Aler carpinfolium 10.0424.19 20.0792.02 30.0793.66 40.0633.560.161 50.0630.72 60.1035.24 70.0421.16 80.0421.12 This table over two pages shows that terminal portions of branches had similar bending stresses as expressed by probability values.

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Species Branch Number Smallest Section Modulus Largest Section Modulus Probability Value Betula pubescens 10.1966.2 0.645 20.1194.18 30.9657.07 40.4478.57 Crataegus diffusa 10.492.3 0.928 20.311.4 30.492.1 40.494.6 Ulmus procera 11.7127.4 0.601 21.322 31.2121.9 40.6817.3 50.3417.9

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

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SpeciesBranchSM at tipSM at base Stress Probability Acer palmatum A2.0297.0 0.33 B0.4781.7 Acer palmatum A0.4718.9 0.9 B2.0236.8 Ailanthus altissima A18.589.7 0.036 B5.1516.2 C12.747.3 Aler carpinfolium A0.04252.5 0.32 B0.04257.9 C0.04231.6 D0.07942.8 Arbutus unedo A1.0925.9 0.87 B1.0946.6 (SSPS Institute Inc, 2000) This table over three pages shows that secondary portions of branches had similar bending stresses as expressed by probability values.

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SpeciesBranchSM at tipSM at baseProbability Betula albosinesis A0.7950.6 0.744 B0.310.8 C0.3331.5 Broussonetia papyrifera A0.9587.7 0.301 B0.9787.4 C0.8597.4 Cedrus libani A1.1739.2 0.31 B1.4670.2 C0.8577.5 Cornus florida A1.56172.3 0.582 B0.38216.3 Liquidamber styraciflua A0.86166.4 0.777 B1.14121.1 C1.56160.2 Malus hypnhesis A8.23179.9 0.819 B14.5185.2

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SpeciesBranchSM at tipSM at baseProbability Phillyrea latifolia A2.0928.03 0.817 B2.0932.34 Phillyrea latifolia A2.0986.7 0.761 B2.0951.8 Pinus thunbergii A1.640.8 0.184 B1.521.5 C1.528.5 Prunus serotina A0.4226.1 0.338 B0.5211.5 C0.4720.9 D5.627.4 Quercus palustrius A1.9101.9 0.029 B2.6108.1

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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.

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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.

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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: 241-253 Castera P, Mortier V. (1991) Growth patterns and bending mechanics of branches Trees 5: 232- 238. 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: 559-568 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: 365-371. 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. 2000. Systat 10 [computer program]. SPSS Institute, Chicago, IL. Wilson B, Archer R (1979) Tree design: some biological solutions to mechanical problems. Bioscience 29: 293-298 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

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