A SCALING LAW FOR THE EFFECTS OF ARCHITECTURE

Transcription

1 Aerican Journal o otany 95(1): A SCALIG LAW FOR THE EFFECTS OF ARCHITECTURE AD ALLOMETRY O TREE VIRATIO MODES SUGGESTS A IOLOGICAL TUIG TO MODAL COMPARTMETALIZATIO 1 Mathieu Rodriguez,,3 Eanuel de Langre, and runo Moulia 3,4 Departent o Mechanics, LadHyX, Ecole Polytechnique-CRS, 9118 Palaiseau, France; and 3 UMR547 PIAF, IRA, Univ laise Pascal, F Cleront Ferrand Wind is a ajor ecological actor or plants and a ajor econoical actor or orestry. Mechanical analyses have revealed that the ultiodal dynaic behavior o trees is central to wind tree interactions. Moreover, the trunk and branches inluence dynaic odes, both in requency and location. ecause o the coplexity o tree architecture, inite eleent odels (FEMs) have been used to analyze such dynaics. However, these odels require detailed geoetric and architectural data and are tree-speciic two ajor restraints or their use in ost ecological or biological studies. In this work, closed-or scaling laws or odal characteristics were derived ro the diensional analysis o idealized ractal trees that sketched the ajor architectural and alloetrical regularities o real trees. These scaling laws were copared to three-diensional FEM odal analyses o two copletely digitized trees with axial architectural contrast. Despite their sipliying hypotheses, the odels explained ost o the spatioteporal characteristics o odes that involved the trunk and branches, especially or sypodial trees. These scaling laws reduce the tree to (1) a undaental requency and () one architectural and three bioetrical paraeters. They also give quantitative insights into the possible biological control o wind excitability o trees through architecture and alloetries. Key words: alloetry; bioechanics; diensional analysis; dynaics; requency; odel; scaling; tree architecture; vibrational properties; wind. Wind tree interaction is a ajor concern or the anageent o orest and urban trees because windthrow and windbreak result in substantial econoical costs and potential huan risks ( Gardiner and Quine, 000 ; Jaes et al., 006 ). Moreover, echanosensing by trees o wind-induced strains ( Coutand and Moulia, 000 ) and induced thigoorphogenetic responses are undaental issues in understanding how trees can control their susceptibility to wind hazard ( Moulia et al., 006 ) and accliate to their wind cliate ( r ü chert and Gardiner, 006 ). Pioneer work on wind tree interactions only considered static deorations under wind load (see review in Moulia and Fournier-Djibi, 1997). Over the last decades, tie-dependent dynaic eects have been ound to play a ajor part in wind deorations and windbreaks (e.g., Mayer, 1987 ; Gardiner et al., 000 ). However, the dynaic interactions between wind and trees are coplex issues ( iklas, 199 ). Wind velocity has a large spectru o eddy size and requency, as well as ean vertical proiles ( de Langre, 008 ). Most trees also have a branched architecture with dierent odes o branching (onopodial vs. sypodial) depending on species, up to 11 orders o axes, and reiterated patterns o various sizes and positions 1 Manuscript received 7 May 008; revision accepted 9 Septeber 008. The authors dedicate this article to the late Dr. H. Sinoquet, who pioneered the use o 3D digitizing and virtual plants or physical issues related to plant biology and who let us too early. They thank Drs. H. Sinoquet and D. Sellier or access to the digitized tree databases and. Dones or teaching the basics o MTG handling and o tree architecture. Thanks also to Dr.. Roan or pointing out the siilarities in the requency ranges o very distinct trees and the two anonyous reviewers and Dr.. E. Hazen or providing helpul suggestions to revise the anuscript and iprove the English. This work was supported by AR grant AR-06-LA Chêne-Roseau. 4 Author or correspondence (e-ail: oulia@cleront.inra.r) doi:10.373/ajb ( artheley and Caraglio, 007 ). Thereore, the developent o echanical odels or wind tree interactions is necessary in addition to experients, and the siplest relevant odel is to be sought (see review in de Langre, 008 ). A central question to be investigated is then obviously the inluence o branched architecture and tree geoetry on the dynaics o trees and the potential biological control o tree resistance to wind through the orphological developent o the tree. The echanical response o a tree to turbulent wind results ro the interaction o three coponents: (1) the luctuating excitation load by wind drag, () the dynaic elastic behavior o the syste, and (3) the daping processes (e.g., Moore and Maguire, 008 ). The load is the input o echanical energy in the syste. The oscillatory elastic behavior o the structure is driven by the conservative exchange between two ors o the internal echanical energy: (1) kinetic energy (the su over all the aterial eleents o the tree o the products o their asses ties their square velocity) and () elastic-strain potential energy ( Gerardin and Rixen, 1994 ). These internal energetic exchanges can be characterized by considering the natural ree-otion dynaics o the isolated syste (i.e., with suppressed energy input and negligible dissipative losses) and by studying the resonance requencies, which characterize the exchange rate between these two ors o energy. Last, the daping processes dissipate a part o the echanical energy out o the structure, thus resulting in the decay o the aplitude o the oscillation. This dissipation involves (1) the production o sall-scale turbulence in the wake through aerodynaic drag at the level o the lea air and branch air interaces and () to a lesser extent in ost cases, the production o heat through internal viscosity (i.e., internal riction) in the wood ( iklas, 199 ). All these coponents are coplex spatioteporal processes. However, a very eicient approach to tackle this coplexity is to ocus irst on the oscillatory elastic behavior o the structure (or exaples in trees, see Sellier et al., 006 ; Moore and Maguire, 008 ). Indeed, the oscillatory elastic

2 154 Aerican Journal o otany [Vol. 95 behavior o the tree represents its intrinsic echanical excitability ( Gerardin and Rixen, 1994 ). Moreover, this oscillatory elastic behavior can be analyzed as the superposition o distinct odes o deoration through odal analysis, a standard tool in echanical engineering. Each ode, nubered j, is an eigen way o oscillatory exchange between kinetic energy and elastic-strain potential energy ( Gerardin and Rixen, 1994 ) in the absence o any daping or external load. The ode j is deined by the odal deoration Φ j (the displaceent vector ield deining the shape o the deored tree), odal requency j, and odal ass j (which characterizes the inertia o the ode). Such odal analysis has two advantages. Modes can be ranked according to their contribution to the overall oveent, so that sipliied odels casting to a liited nuber o odes can be deined objectively. Additionally, both the excitation load and the dissipation processes can be projected onto the set o odes (the odal basis) and thereby analyzed. For exaple, soe odes will be ore excited than others by a given load i this load applies to places where the odal deoration induces large displaceents, even ore i this excitation occurs at requencies close to the corresponding natural odal requencies. Such increased excitation is due to the phenoenon o resonance that results in a rapid uptake o the energy by the oscillating syste with attendant apliication o the aplitude o the corresponding ode. y the sae token, i large and ast oveents o the branch tips occur in these odes, they will be ore daped than others through aerodynaic daping. Modal analysis is thus a useul prerequisite to a ore coplete dynaic odeling o wind tree interactions. A ew authors have used odal analysis on trees (e.g., Fournier et al., 1993 ; Sellier et al., 006 ; Moore and Maguire, 008 ). All have concluded that odes involving signiicant branch deoration could rank in between odes deoring ainly the trunk ( Fournier et al., 1993 ; Sellier et al., 006 ). Experients ro Moore and Maguire (005) and Sellier and Fourcaud (005) conired the excitation o several odes in conier trees under wind load, with again soe o the odes having their deoration ainly located on branches. Although not using odal analysis, Jaes et al. (006) also showed that the easured requency spectra o the responses under wind excitation o our trees with dierent architectures, including coniers, two eucalypts and a pal tree, were also signiicantly dependent on the branching syste. Moreover, Jaes et al. (006) and Spatz et al. (007) argued that such ultiodal dynaics including branch deoration could be beneicial to the tree by enhancing aerodynaic dissipation through a echanis called ultiple resonance daping or ultiple ass daping. All these works contrasted clearly with previous works in which the dynaic contribution o branches and oliage was reduced to that o luped asses ixed on a lexible bealike trunk (e.g., Gardiner, 199 ; Spatz and Zebrowski, 001 ) and where only the irst and eventually second odes o deorations o the trunk (i.e., irst-order axis) were considered, both theoretically and experientally. ote, however, that only onopodial architectures ( Gillison, 1994 ; artheley and Caraglio, 007 ) were considered in all these studies about branch dynaics and that the only nonconier trees were eucalypts. A coparison with sypodial trees would be interesting beore generalizations are ade. In view o such coplex eects, it ay see that only detailed inite eleent odels (FEM) o the three-diensional (3D) architecture o trees can be used or wind tree interaction studies o coplex branched architecture and that only siulation studies can be done ( Sellier et al., 006 ; Moore and Maguire, 008 ). Such siulation studies require a speciic odel or each individual tree, with intensive 3D descriptions o tree architecture ( Sinoquet and Rivet, 1997 ; Moore and Maguire, 008 ), while providing a liited perspective or generality and or the analysis o possible biological control. The geoetry o trees, however, has in ost cases soe architectural syetry related to the branching pattern (onopodial vs. sypodial growth) and spatial bioetrical regularities such as the alloetry law or slenderness, which relates length and diaeter o segents ( McMahon and Kronauer, 1976 ; iklas, 1994 ; Moulia and Fournier-Djibi, 1997). The geoetry o a tree can thus be approxiately suarized using a ew paraeters. Moreover, the setting o soe o these paraeters is controlled through thigoorphogenetic processes ( Moulia et al., 006 ). ut the issue o whether such regularities can be relected in general scaling laws or odal characteristics o sypodial and onopodial angiosper and/or conier trees has not been addressed yet. The hypothesis here is that siple scaling laws should occur. I they were to exist, these laws would ake the studies o tree dynaics easier (ethodological aspect) and give insights into the possible biological control o the overall tree dynaics excitability through genetic or thigoorphogenetic changes in their paraeters. The ais o the present paper are thus (1) to explore the respective role o the architecture and alloetry paraeters on odal characteristics by cobining FEM odeling and diensional analysis o an angiosper tree and a conier tree with highly contrasting sizes and architectures and () to assess whether ore generic scaling laws relating tree ultiodal dynaics and architectural and geoetrical paraeters can be deined and to discuss their biological signiicance. In the irst section, the odal characteristics o two 3D odels o extensively digitized trees with very distinct architectures one sypodial angiosper (walnut) and one onopodial conier (pine) are analyzed. Then in the second section, idealized ractal tree odels are deined to explore the inluence o bioetrical regularities and branching patterns on the odal characteristics o a tree. Scaling laws between the successive odes o a tree based on the global paraeters characterizing architecture and slenderness are then derived in these idealized trees. Finally, in the third section, we show that walnut and pine odal characteristics can be approxiated using these scaling laws, so that the general ultiodal behavior o trees can be reduced as a irst approxiation to (1) the irst ode and to () general scaling laws or higher odes. MODAL AALYSIS OF A WALUT AD A PIE TREE MATERIALS AD METHODS 3D descriptive data Two real trees with highly contrasting sizes and architectures were considered ( Fig. 1 ). The irst tree, Fig. 1A, is a 0-yr-old walnut tree ( Juglans regia L.) described in Sinoquet et al. (1997). It was 7.9 high, 18 c in diaeter at breast height (dbh), and had a sypodial branching pattern and eight orders o branching. The second tree, Fig. 1, is a 4-yr-old pine tree ( Pinus pinaster Ait.) described in Sellier and Fourcaud (005). It was.6 high, 5.6 c in diaeter at 13 c height, and had a onopodial branching pattern and three orders o branching. The geoetries o these two trees (positions, orientations, diaeters o the ste segents, and the topology o branching points) are known in great detail through 3D agnetic digitizing ( Sinoquet and Rivet, 1997 ) and are organized

3 Deceber 008] Rodriguez et al. Tree vibration odes 155 Fig. 1. Geoetries o (A, C) a walnut tree (ro Sinoquet et al., 1997 ) and (, D) a pine tree (ro Sellier and Fourcaud, 005 ). in databases using the Multiscale Tree Graph MTG structure ( Godin et al., 1999 ). ecause our central concern was on the eects o branch architecture and alloetry, these two trees were analyzed in this study without considering leaves or needles. FEM odeling and coputation o odal characteristics In slender structures such as trees, the bea theory applies ( iklas, 199 ), and odal deorations involve ainly bending and torsion. The echanical odel used or both trees were thus based on representing each branch segent as a bea, described by its lexural stiness and inertia, using the Euler theory o linear elastic beas ( Gerardin and Rixen, 1994 ). ea sections were assued to be circular, with a variable diaeter along the bea (taper) when available. Connections between branch segents were set as rigid. The root anchorage was odeled as a perect claping condition at the tree basis. The green-wood aterial properties (density ρ, Young odulus E, and Poisson ratio ν ) were assued to be unior over all branches o each tree. Their values ( Table 1 ) were taken ro the easureents in Sellier et al. (006) or the pine tree, while or the walnut tree, their values were estiated using Eq. 1 in Fournier et al. (005). For each tree, a inite eleent representation was built using the CASTEM v. 3M sotware ( Verpeaux et al., 1988 ). The stiness and the ass atrices o the inite eleent orulation were then coputed. y solving the equations or ree otions using these atrices, we can ully deine the odes (requency, shapes, and ass) (see also Fournier et al., 1993 ; Moore and Maguire, 005 ; Sellier et al., 006 ). ecause odes o ree otion only involve exchange between kinetic energy and elastic strain en- ergy, they can thus be studied without considering gravity eects and are deined under the assuption o linear transoration (sall displaceents, sall strains). RESULTS In the walnut tree odel ( Fig. A ), the irst 5 odes were ound in the range between 1.4 and.6 Hz. A sall but clear requency gap (~ +0.4 Hz) occurred between the irst two odes and the ollowing ones. Then odal requencies continued to increase with ode nuber but at a saller rate. Table 1. Geoetrical and echanical characteristics o the two analyzed trees (walnut, Juglans regia ; pine, Pinus pinaster). Species Age (yr) Height () Diaeter ()ρ [kg 3 ] E [GPa]ν J. regia (dbh) P. pinaster (at 13 c) otes: ρ, wood density; E, Young odulus; ν, Poisson ratio

4 156 Aerican Journal o otany [Vol. 95 Fig.. (A) Mode requencies o the walnut tree classiied in ters o ain localization o bending deorations: I, in the trunk; II, in second-order branches; III, in third-order branches. () Mode requencies o the pine tree classiied in ters o ain localization o deorations: I, irst bending ode o the trunk; I, second bending odes o the trunk; II, bending odes in second order branches. (C) Sketches o the ode deorations in the case o the walnut tree. (D) Sketches o the ode shapes in the case o the pine tree. These odes can be classiied according to their odal deoration Φ j ( Fig. C ). The irst group o odes, labeled I, displayed a bending deoration ainly in the trunk basis. This resulted in a lateral displaceent o the upper part o the bole ostly through rigid-body rotational eect, as sketched in Fig. C. In other words, deorations occurred ostly in the trunk up to the crotch, and the branches swayed like rigid bodies. Group I included two odes, corresponded to the bending in the x and y direction, respectively, with identical requencies o ~1.4 Hz. The second group, labeled II, corresponded to deorations ainly located on irst-order branches, with ostly rigid-body displaceents o the branches o higher orders. ecause there were six ain branches bending in the x and y directions, respectively, this second group included 1 distinct odes, each one with dierent contributions o the deoration o these branches (odes 3 14). The third group, labeled III, corresponded to odes 15 5 with deorations ainly localized on thirdorder branches. In the case o the pine tree ( Fig. ), the irst 5 odes were ound in the range between 1 and.8 Hz with, here again, a clear sall gap between the irst two odes and the ollowing ones. The group o the irst two odes, labeled I, had requencies at 1.08 Hz, identical to coputational results ro Sellier et al. (006) on the sae tree ( Table ). ending deorations o these odes are ainly located in the trunk with displaceents in the whole branching syste. A second group o odes, labeled I, involved signiicant deoration in the trunk, but displayed higher odal requencies around.41 Hz (odes 17 and 18, Fig. ). This group corresponded to the second bending odes o the trunk with bending deorations spread over the whole tree ( Fig. D ). As in the walnut tree, the two other groups o odes, labeled II and III, were associated with deorations located ainly on second- and third-order branches, respectively. For both trees, odal requencies were concentrated in a sall requency range. With requencies o 1 Hz order o agnitude and ~10 odes per Hz, the requency spacing o the odes was typically 0.1 Hz. Such a high density o odal requencies is consistent with the conclusions o Jaes et al. (006) and Spatz et al. (007). ote, however, that the organization o odes groups diers between the two trees: the sequence is I / II / III or the walnut and I / II / I / II or the pine tree.

5 Deceber 008] Rodriguez et al. Tree vibration odes 157 THEORETICAL COSIDERATIOS: SCALIG LAWS I IDEALIZED TREES Idealized ractal trees To explore the respective eects o tree architecture and alloetry on odal characteristics, we deined idealized ractal trees. Although not copletely realistic (e.g., very short internodes are neglected so that several branches can be inserted at the sae branching point, and no axis dierentiation is considered), ractal tree construction is the siplest way o generating a reiterated architecture, with sel-siilar reiterations diering only by scaling coeicients ( Prusinkiewicz and Lindenayer, 1996 ). Two dierent ractal odels were built, representing two extree botanical branching patterns in the existing architectural odels o plants ( artheley and Caraglio, 007 ) The irst odel tree is inspired ro the Leeuwenberg architectural odel (e.g., cassava) and will be reerred to as the sypodial tree in the ollowing. At each branching point, the sypodial tree has syetric lateral segents and no axial segent ( Fig. 3 ). The second odel tree is a highly hierarchical tree inspired ro the Rauh architectural odel (e.g., pine) and will be reerred to as onopodial. It has an axial segent and lateral segents at each branching point ( Fig. 3 ). In both idealized odels, lateral segents have equally spaced aziuthal directions. Tree branches were indexed using the nubers o lateral and axial branching upstrea in the direction o the tree base ( Fig. 3 ). A branch segent in a onopodial tree is indexed [,P ] i this segent has 1 lateral and P 1 axial upstrea (i.e., ore basal) branch segents. In the sypodial idealized tree, the sae syste holds, but P is constant and equal to 1. Thus a single index can be used and a branch segent o a sypodial tree is indexed when it has 1 lateral upstrea branch segents. Segents sizes were deined using three paraeters ( Fig. 3 ): (1) the slenderness coeicient β, corresponding to an alloetric law or branch segents ( Fig. 3C ); () the lateral and axial branching ratios λ and μ, which deine, respectively, the ratio between cross-sectional areas o segents ater and beore branching ( Fig. 3D ); and (3) the angle α o divergence o lateral branches ro the axial direction o the parent segent ( Fig. 3E ). The ean slenderness o the population o branch segents o both trees was thus described using an alloetric law that relates the length L and the diaeter D o each segent ( Mc- Mahon and Kronauer, 1976 ), in the or D ~ L β, (1) whereas the successive diaeters at branching points were D + 1 = μd ( P, ) ( P, ) (A) and D( + 1, P) = λ D(, P). () ote also that the particular case where λ + μ = 1 (i.e., the total section beore and ater branching are identical) corresponds to Da Vinci s surace conservation law ( Prusinkiewicz and Lindenayer, 1996 ), but this was not specially assued hereater. The algorith or generating a ractal tree as in Fig. 3 was the ollowing: (1) the ost basal diaeter and the initial growth direction is given; () using the segent slenderness alloetry (Eq. 1), the length o the irst segent is coputed; (3) a irst branching occurs, and the diaeter o each lateral branch is coputed using Eq. or a sypodial tree or Eqs. A and or a onopodial tree; and (4) their spatial inclination is controlled by the angle α o divergence ro the actual inclination angle o the parent axis and equal aziuthal spacing. For each branch, the process is iterated ro step (1). In a theoretical ractal, this recursive iteration is ininite. ote however that this algorith or tree construction is not a proper description o the real architectural developent and axis growth o real trees ( artheley and Caraglio, 007 ). For exaple, in real botanical trees, axes (e.g., branches) can usually be deined as a succession o segents (starting at a given branching point [,P ]) that are integrated by cabial growth ( artheley and Caraglio, 007 ). These idealized trees are just designed as a tool to sketch soe o the ajor syetries and possible scalings o trees and to test the against real trees. In the recursive process o construction o the idealized trees, the size o the successive segents depends on the segent slenderness alloetric coeicient, the ode o branching (sypodial vs. onopodial), and the coeicients o area reduction at a branching point. For the sypodial, idealized tree odel ( Fig. 4A ), the relation between successive segents is L L L = λ β and D 1 D L 1 = λ ( 1) β and D D + = 1 1 λ, so that ( 1) = λ. (3) An axis length, l, is deined as the su o the lengths o the segent L and all successive segents ollowing a path o lateral branching, giving 1 l = L = L. (4) 1 = 1 λ β According to Eq. 4, the slenderness coeicient, β, linking a segent length to its diaeter, also links an axis length to its diaeter: D = k1l = kl β β. In the case o the odel tree o onopodial type ( Fig. 4 ), the scale o a segent depends on the position o the parent segent in the central onopodial axis. I 1 is the nuber o lateral branching and P 1 is the nuber o axial branching, the relation between diaeters is given by L L P, 11, D ( 1) ( P 1) β = λ μ β and D P, 11, 1 P 1 = λ μ. (5) Due to the slenderness alloetry, the reduction in segent 1 length and segent diaeter are thus linked by rl = r β D. Axis length, l, P, is deined as the su o the lengths o the segent L, P and all its son segents through axial branching, giving P, P, P, P = P l = L = L 1 1 μ 1β. (6) According to Eq. 6, the slenderness coeicient, β, linking a segent length to its diaeter, also links the axis length to its basal β diaeter: D = k L = k l β. P, 1 P, P,

6 158 Aerican Journal o otany [Vol. 95 Table. Frequencies o walnut as described in Sinoquet et al. (1997) and o pine tree as described in Sellier et al. (006), ound via a inite eleent analysis. Walnut ( Juglans regia L. ) Mode nuber Group Mode requency (Hz) Mode nuber Group Pine tree (Pinus pinaster Ait.) Mode requency (Hz) Sellier et al. (006) Present results Exp. Cop. 1 I I I 1.41 I II II II II II II II II II II II II II.01 9 II II.0 10 II.1 11 II II.1 1 II.06 1 II.3 13 II II.4 14 II.1 14 II.7 15 III.7 15 II III.9 16 II.3 17 III I III I III.4 19 II.58 0 III.44 0 II.60 1 III.47 1 II.66 III.48 II.69 3 III.51 3 II.70 4 III.54 4 II.7 5 III.56 5 II.7 Finally, in the idealized ractal trees, sel-siilar subsets o the tree starting ro any branch biurcation ay be identiied, as illustrated in Fig. 4. These subsets can thus ollow the sae index as their basal segent (i.e., or, P ). ecause o the assuption o the reiterated sel-siilar branching law, a given subset is identical to the whole tree, except or its ain axis length scale, l or l, P, and its diaeter scale d or d, P. General scaling laws or odal characteristics In a echanical odel o a syste in which segents are represented as beas o circular sections, two length scales exist. The irst one, l, ixes the scale o coordinates o these segents. A second one, d, which scales the diaeters o these segents, is needed. In general, these two scales are not related because a given geoetry o segents ay correspond to several characteristic diaeters d. Moreover, we can assue that aterial properties o the wood (density ρ, Young odulus E, and Poisson ratio ν ) are constant within the tree and that their possible dependence on l and d can be neglected. The relation between odal requencies and these two scales ay then be assessed by standard diensional analysis. A odal requency depends on lengths, scaled by l, on asses per unit length, scaled by ρ d, and on the bending stiness k, scaled by Ed 4, and is written as F l, ρd, Ed 4. (7) = ( ) ut, because a physical law is by nature independent o units, this relation ust be expressed in ters o diensionless paraeters ( iklas, 1994 ; Chakrabarti, 00 ). Considering the respective diensions o these our variables ( ~ Tie 1, ρ d ~ Mass Length 1, Ed 4 ~ Mass Length 3 Tie ), the diensional equation corresponding to Eq. 7 reads ( ) ( ) = 1 1 b 3 c a b+ 3 c b+ c c a T = L M L M L T L M T, (8) yielding c = 1/, b = 1/, a =. All our variables ay thus be cobined in a diensionless paraeter: ( ) ( ) l d 1 Ed 4 1 ρ / /, (9) iplying that the relation expressed in Eq. 7 is o the or Or equivalently, ( ) ( ) = 1/ 4 1 / l ρd Ed E ~ l ( d ) ( Ed ) ~ l d 1 / 4 1 / ρ ρ constant. (10) 1 /. (11) Assuing that the odal ass depends on the ass per unit length, scaled by ρ d, and the length, scaled by l, the diensional analysis yields ~ ld. Siilarly, the odal stiness, k, deined ro π = k, scales as k ~ l 3 d 4. In a given tree, where an alloetric law relates length and diaeter o each segent, d/l β = constant, i.e., d ~ l β. Then,

7 Deceber 008] Rodriguez et al. Tree vibration odes 159 Fig. 4. Identiication o subsets in (A) the sypodial and () the onopodial odel trees. Subsets are circled in black or gray. Fig. 3. Exaples o odel ractal trees and paraeters deining odel trees. (A) Sypodial case, α = 1.5, and () onopodial case, α = 30 ; (C) ranch slenderness coeicient, β ; (D) branching ratios, λ and μ ; and (E) angle o branching connections, α, illustrated here in the case o two lateral branches. ro Eq. 11, the requencies o odes are expected to depend on the scale o length and o diaeter as ollows: ~ l β ~ d β β and siilarly or the odal ass and stiness: 1+ b ~ l 1 + b b 4b 3 ~ d, k~ l ~ d 4b 3 b (1). (13) Relation between requencies or a ractal tree Due to the syetries o the ractal structure, groups o odes can be deduced and classiied according to their odal deoration ( Fig. 5 ). Soe odes involve trunk deoration (group I in sypodial tree [ Fig. 5A ], group I,I in the onopodial tree [ Fig. 5 ]). Other odes involve ainly the bending deoration o the basal branch o all subsets o the sae order (e.g., odes II or = subsets, odes III or = 3 subsets in the sypodial tree [ Fig. 5A ], ode II,I or [,1] subsets and II,II or [,] subsets in the onopodial tree [ Fig. 5 ]) with negligible deoration o upstrea segents. The deoration o upstrea segents is strictly zero when the ode involves the syetric deoration o two syetric subsets as in Fig. 5. In the odes where syetric subsets are deored antisyetrically, the lower part o the tree is slightly bent. ut the elastic strain energy stored in this slight bending o the lower part o the tree is negligible copared to the energy stored in this sae part due to odes with lower index. Consequently, scaling laws will be derived thereater con- sidering only odes o syetric deorations o subsets. As the general bioetrical laws apply to subsets, the hypothesis to be tested is that the scaling laws derived ro syetric odes will capture the diensional behavior o the whole group o odes involving the deoration o all the subsets o a given scale. For a sypodial, idealized tree ( Fig. 5A ), the odal deorations o three groups o syetric odes (I, II, and III) can easily be deduced ro one to the other. ecause o the syetry o the branching pattern, a ode o group II is associated with the deoration o a subset with a ixed part at its base. Thereore, the odal requency o the group II o the whole tree can be considered as the requency o a ode o group I o the subset i it is isolated. The dependence o II on l II and d II should thereore be identical to the dependence o I on l I and d I, hence yielding: II I dl II II dii = = dl I I di β β. (14) Using the relation between successive diaeters in a ractal sypodial tree, Eq. 3 yields II I β β = λ. (15) Siilarly, the requency o odes in the group o order is given by I = λ ( 1) ( β ) β. (16) Thereore, all requencies can be deduced ro the irst one, given the alloetric paraeter β, and the area reduction paraeter at branching λ. In the case o the odel tree o onopodial type ( Fig. 5 ), the scale o a subset [,P] depends on its central axis length and diaeter, l, P and d, P. Introducing the relation between diaeters and between lengths ro Eq. 5 in Eq. 1, the corresponding requency ratio can be deined as P, 1 P 1 I,I = λ μ β β. (17)

8 1530 Aerican Journal o otany [Vol. 95 Table 3. Slenderness coeicients β, lateral and axial branching ratios μ and λ, respectively, and branching angles α used to illustrate the two idealized trees. A slenderness coeicient equal to 3/ has requently been used in the literature (see McMahon and Kronauer, 1976 ; Moulia and Fournier-Djibi, 1997), and branching ratios were chosen to ollow Da Vinci s surace conservation law ( Prusinkiewicz and Lindenayer, 1996 ). Idealized tree β λ μ α Sypodial 3/ 1/ 0 0 Monopodial 3/ 1/6 /3 30 Fig. 5. (A) Modes o groups I, II, and III o the sypodial odel tree. () Modes [I,I], [II,I] and [II,II] o the onopodial odel tree. ( ) and ( ) represent the centers o bending energy and kinetic energy respectively. Through siilar arguents, the odal ass o the groups o order and [,P ] or a sypodial tree and a onopodial tree read, respectively, as I = λ ( 1) ( 1+ β) β, P, 1 P 1 I,I = λ μ 1+ β β. (18) Center o bending energy and center o kinetic energy Coparing the spatial distributions o odal displaceents Φ j along the tree is not straightorward. To suarize the localization o displaceent associated with a given ode, one ay deine two geoetrical points: the center o kinetic energy o the ode and the center o elastic bending strain energy o the ode. The odal center o kinetic energy is located at an elevation z K such that z K ρsφφ dω = ρsφφ zdω. (19) Ω Siilarly, the odal center o bending strain energy, z, is deined by a vertical position Ω z EIγ γdω = EIγ γ zdω, (0) Ω where γ is the curvature associated with the odal displaceent Φ and where integration is perored over the whole tree ( Ω ). These two paraeters scale, respectively, as z K ~ l ~ d 1 / β and z ~ l ~ d 1 / β. Exploiting again the assuption o reiterated trees, the position o the centers or a ode [,P ] reads Ω K ( P 1) β 1/ β 1 K z z λ cos α z, (1) = +μ ( ) P, 1, P ( P 1) β 1/ β 1 z z λ cos α z, () = +μ ( ) P, 1, P where z, P is the elevation o the branching biurcation (see Appendix 1 or the geoetrical derivation). I,I I,I uerical illustration o scaling laws on idealized ractal trees The scaling laws derived in the preceding section were applied to two particular occurrences o the idealized trees, one sypodial and one onopodial. The alloetric and geoetrical paraeters deining their geoetry are given in Table 3. Modal requencies Instantiating the values o the alloetric and geoetrical paraeters in Eqs. 16 and 17 using Table 3, the series o requencies or the sypodial [ ] and onopodial [,P ] idealized trees read, respectively, as I = 1 6, P, I,I = P 1 6. (3) These series o requencies are illustrated in Fig. 6A and 6. In the case o the sypodial [ ] idealized tree in Fig. 6A, requencies o group o odes are seen to increase progressively. Conversely, or the onopodial [,P ] idealized tree in Fig. 6, sets o requencies corresponding to the double-indexed branching pattern can be observed. For a given value o (i.e., or sequential subtrees along a given onopodial axis, e.g., = ), requencies increased progressively with P. For a given P (i.e., or series o lateral subtrees, e.g., P = 1), requencies also increased with. Fro Eq. 3, it appears that, P values ro dierent groups intercalate, e.g.,,5 < 3,1 <,6. In the two cases, however, the organization o requencies is clearly dependent on the architecture, through the paraeters λ and μ o area reduction at branching, and on the slenderness alloetry, through the paraeter β. y the sae token, the odal ass reads I = 4 ( 1 ) 3, P, I,I 4 P = 4 ( ) ( 1) (4) Localization o odal ass and odal stiness The localization in height o the centers o bending energy and the centers o kinetic energy or the odes in each idealized tree were deterined ro Eqs. 1 and using the values or paraeters ro Table 3, then plotted as a unction o the corresponding odal requency ( Fig. 6C, D ). In both trees, the distance between the center o bending energy and the center o kinetic energy is a decreasing unction o the odal requency. Modes thus tended to be ore local as their odal requency increased. In the sypodial tree in Fig. 6C, odes localized higher in the tree as the odal requency increased. In the onopodial tree in Fig. 6D, the ixed axial and lateral branching pattern

9 Deceber 008] Rodriguez et al. Tree vibration odes 1531 Fig. 6. Vibration odes o the sypodial and onopodial idealized trees. Modal requencies, relative to the irst one, as a unction o the index o the corresponding subsets in the case o the (A) sypodial and (C) onopodial trees, respectively. Vertical position o the centers o bending energy ( - ) and o the centers o kinetic energy ( - ) as a unction o the requency, respectively in the (C) sypodial and (D) onopodial trees. resulted in the centers o bending energy and the centers o kinetic energy being scattered all along the trunk. Modes related to group o subtrees localized at a constant nuber o lateral branching (ixed, changing P ) were ound to be localized higher in the tree as the odal requency increased. ut the intercalation o odes both in ters o spatial localization and requency is obvious. The odal analysis o idealized trees thus suggests that the localization o odes in the structure depends o the tree architecture via its branching pattern (sypodial vs. onopdial). TEST OF THE SCALIG LAWS O MODELS OF REAL TREES The scaling laws or odal requencies (Eqs. 16 and 17) and localization o the centers o kinetic and bending energy (Eqs. 1 and ) were then applied to the two real tree odels (i.e., the sypodial walnut and the onopodial pine, Fig. 1 ). The results were then copared with the odal characteristics coputed using 3D inite eleent odels (see section Modal analysis o a walnut and a pine tree ; Fig. ).

10 153 Aerican Journal o otany [Vol. 95 Deterination o bioetrical paraeters or the scaling law Orthogonal regressions (SAS version 9.1, procedure Insight Fit) were applied to estiate slenderness alloetric coeicients, β, and branching ratios, λ and μ, ro the MTG tree databases o the walnut and pine geoetries. Slenderness coeicients, β, were estiated ro the linear orthogonal regression log( d ) = β log(l ) + k ( iklas, 1994 ). Lateral branching ratios, λ, were estiated ro linear orthogonal regression ( d ) = λ (d -1 ) and axial branching ratio, μ (in onopodial tree), ro ( d 1, P ) = μ (d 1, P -1 ). Regression coeicients, root ean square errors, coeicients o deterination, and 90% conidence intervals (Dagnelie, 006) are reported in Table 4. Case o the walnut tree Paraeters were estiated using data ro the irst three order branches with diaeters and lengths larger than 1 c and 1, respectively ( Fig. 7A, ). A highly signiicant, tight alloetric relation was ound between l and d ( Fig. 7A ), capturing 87% o the total variance, with only two outlying points corresponding to the trunk and to a cut branch. The relation between d +1 and d was a little bit ore biased, but still a highly signiicant λ could be deined. The sypodial branching pattern o the walnut iplies an axial branching ratio μ equal to 0. The angle o branching has been ound to vary between 0 and 40, a ean angle o branching, α = 0, was retained. Case o the pine tree Paraeters were estiated using data ro the irst two order branches ( Fig. 7C E ). The slenderness β ( Fig. 7C ) and the longitudinal area reduction μ (Fig. 7E ) were statistically signiicant. The axial branching ratio λ (Fig. 7D ) was also signiicantly dierent ro zero, but the relation between the cross-sectional area o the parent segent and o lateral branches was very poor [the slope o the regression odel with the intercept is not signiicantly dierent ro zero with probability p ( > F ) = 0.13]. The ean angle o branching was α = 30. Application o scaling laws Using the paraeters corresponding to the two real trees ( Table 4), we applied the scaling laws derived in the Theoretical considerations section to the case o each real tree, then copared the results to the odal characteristics coputed using the 3D inite eleent odels Table 4. Slenderness coeicients β, lateral and axial branching ratios μ and λ, respectively, and branching angles α estiated ro walnut and pine tree geoetries (orthogonal regression coeicients, conidence intervals at 90% level [CI), coeicients o deterination [ R ], and root ean square o the residual errors [ σ res ]). Tree geoetries are ro Sinoquet et al. (1997) and Sellier and Fourcaud (005), respectively. ote that or λ and μ, the regressions were obtained with no intercept so that the R value cannot be copared directly with that related to a standard linear regression, and signiicance levels are to be related to a null hypothesis where the dependant variable is equal to zero (or a ore detailed discussion, see Freund and Littel, 1991 ). Tree β λ μ α Walnut CI 1.5 <β < <λ <0.9 R σ res Pine CI 1.5 <β < <λ < <μ <0.79 R σ res deined in the section Modal analysis o a walnut and a pine tree ( Fig. 8 ). Case o the walnut Figure 8A displays the requencies o the three groups o odes (I, II, and III, see Fig. A, C and section Modal analysis o a walnut and a pine tree ) estiated using the 3D FEM vs. the group nuber. On the sae graph, the dotted lines show the requencies predicted using Eq. 16 with the 90% conidence range o paraeters as in Table 4. The sae coparison is held or the values or the height o the bending and kinetic energy centers (using Eqs. 1 and ), in Fig. 8 and C. Though the geoetry o the real walnut is uch ore coplex than that o the idealized ractal sypodial tree, the prediction using the scaling laws quite closely brackets the range o odal characteristics o the FEM odel. The positions o the centers o bending and kinetic energy are particularly well estiated, with the exception o two points. Case o the pine As ephasized previously, the case o the onopodial tree is uch ore coplex, due to the double index dependence related to the two kinds o branching, axial and lateral. We will ocus on the characteristics o the odes labeled II in the section Modal analysis o a walnut and a pine tree (see Fig., D ), corresponding to the otion o lateral subsets. In ters o the double index reerence, these are odes involving [, P ] subsets. The scatter o odal characteristics is higher in the pine tree than in the walnut tree ( Fig. 8D F ). The scaling law derived ro idealized ractal onopodial tree still brackets ro 60 to 75% o the outputs o the FEM odel. ut hal o the conidence intervals ro the scaling laws does not contain any output o the FEM odel. Moreover, odes o group I cannot be predicted, using the scaling laws applied to subsets o the tree because the corresponding deoration cannot be deined as the deoration o a subset. For instance, the requencies o I are in the order o.4 Hz or the pine tree, while those corresponding to subset [1,] are about 1.16 Hz. DISCUSSIO An approach o the coplex oscillatory behavior o trees through odal and scaling analyses Despite its standard use in echanical engineering ( Gerardin and Rixen, 1994 ), odal analysis has only been used in a ew studies to analyze the dynaic characteristics o trees in relation to their 3D architecture ( Fournier et al., 1993 ; Moore and Maguire, 005 ; Sellier et al., 006 ). Copared to the analysis o the vibrational behavior o separated eleents o the tree such as trunk, branches (e.g., Mc- Mahon and Kronauer, 1976 ; Spatz et al, 007 ), odal analysis takes into account the additional act that as a whole, the tree is a echanical structure. As a consequence, elastic strain energy is alost instantaneously distributed over the whole tree structure, and vibrations involving the whole tree can occur. Such vibrations can involve several parts o the tree together and can thus be ore coplex than that o isolated parts. Indeed, odels connecting a large nuber o sall daped oscillators connected together each oscillator odeling a branch subsyste have been proposed recently ( Jaes et al., 006 ). However, such odels are not parsionious, and their behavior ay be diicult to analyze quantitatively. Under the classical assuption

11 Deceber 008] Rodriguez et al. Tree vibration odes 1533 Fig. 7. ioetrical relations in the walnut and the pine trees. (A, C) Alloetric relations between length and diaeter o branches in (A) the walnut and (C) the pine tree. ( ) Orthogonal regression D ~ L β, with β = 1.37 (A) and β = 1.38 (C). (, D) ranches cross-sectional areas beore and ater a lateral branching in () the walnut and (D) the pine tree. ( ) Orthogonal regression ( d ) = λ ( d -1 ), with () λ = 0.5 and (D) λ = (E) Cross-sectional areas o the trunk beore and ater a branching in the case o the onopodial pine tree. ( ) Orthogonal regression ( d 1, P ) = μ (d 1, P -1 ), μ = Gray areas in the graphs correspond to the 90% conidence level (see Table 4 ). ( ) easured data ro Sinoquet et al. (1997) and Sellier and Fourcaud (005). o relatively sall displaceents, this coplex behavior can be analyzed as the superposition o a (large) set o uch sipler ree-vibration odes with characteristic odal requency and odal deoration and odal inertial ass. These odes are echanical attributes o the whole tree structure, its intrinsic dynaical characteristics independent o any particular load. They characterize the vibrational excitability o a given tree. A given load, say, a turbulent wind with speciic requencies and spatial distributions, will excite only the odes with copatible requencies and odal shapes. ecausee universal wind spectra have been obtained showing that echanically active wind loads in trees typically occur in the 0 10 Hz band ( Stull, 1988 ), and the drag ainly applies to the lea-bearing terinal segents, it is possible to ocus on the subset o odes in this requency band and with odal deorations involving signiicant displaceents o the branch tips, as done in this study. ecause they are based on very detailed and extensive architectural and echanical data, odal analyses can also provide guidelines or deining sipler odels, as illustrated through scaling analysis (and discussed later). eore discussing the ajor insights on tree echanics obtained through this ethod, we should discuss its liitations. In its strict deinition, odal analysis only deals with sall displaceents, as is the case or trees subitted to oderate winds. ut with large winds, large displaceents occur, and geoetric nonlinearities such as strong strealining or branch collisions have to be taken into account ( de Langre, 008 ). Such nonlinear behaviors are still a very active area o research in the echanics o luid structure interactions ( de Langre and Axisa, 004 ). However, soe nuerical ethods or predicting low-induced vibrations in nonlinear cases still involve odal analysis ( Axisa et al., 1988 ), eaning that odal analysis is still a robust starting point or the analysis o the dynaic excitability under strong winds. ranches are iportant to the tree dynaics The detailed FEM odal analysis o entirely digitized trees with a very large contrast in their echanical architecture and odal behavior conired and extended previous reports: odes involving signiicant branch deoration could have requencies very close to and even rank in between odes deoring ainly the trunk ( Fournier et al., 1993 ; Moore and Maguire, 005 ; Sellier and Fourcaud, 005 ; Jaes et al., 006 ; Sellier et al., 006 ). As any as 5 odes could be ound with requencies between one and two ties the ost basal ode involving the trunk and

12 1534 Aerican Journal o otany [Vol. 95 Fig. 8. Coparison o the prediction ro the scaling laws with the inite eleent results on the true tree geoetry: (A) requencies, () centers o kinetic energy ( ), (C) centers o bending energy ( ) o odes o the walnut tree, and (D) requencies, (E) centers o kinetic energy ( ), (F) centers o bending energy ( ) o odes [, P ] o the pine tree. ( and ) are results ro the FEM on true tree geoetries; gray areas are predictions ro the scaling laws, Eqs. 1, 13, 18, and 19, using idealized tree odels. with a typical odal spacing as low as 0.1 Hz, consistent with the results o Jaes et al. (006) and Spatz et al. (007). Although these odes are coplex, they can be classiied using their requencies and odal deoration. In this study, no odes involving an inlection in their odal deoration, I (i.e., second odes o the trunk, I in our labeling), could be observed within the 5 irst odes in the walnut tree, whereas or the pine tree, I only rank 17th and 18th (i.e., 14 odes II involving irst order branches ranked in between the undaental ode o the trunk I and its I ode). This is quite in contrast with clais ro the literature, ostly about adult conier trees, on which only irst I and second I bending odes o the trunk have been reported (e.g., White et al., 1976 ; Mayer, 1987 ; Hassinen et al., 1998 ; Kerzenacher and Gardiner, 1998 ). However, in these studies only the strains in the trunk were easured or odeled; thereore, only odes involving signiicant deoration o the trunk could be recorded. Indeed, when analyzing the vibration odes o an adult aritie pine ( Pinus pinaster ) using inite eleent analysis, Fournier et al. (1993) also ound that odes concentrated in requency and that odes o the second group ranked between the irst and the second bending odes o the trunk. ranch deoration is thus an iportant aspect o trees dynaics whatever the architectures and size (see also Fournier et al., 1993 ; Sellier and Fourcaud, 005 ; Moore and Maguire, 005 ; Spatz et al., 007 ). Scaling laws can be deined As hypothesized, and despite the aoreentioned coplexity o the 3D architecture and odal structure o real trees, scaling laws based on the assuptions o (1) idealized alloetric ractal trees and () syetric odes o branches, are able to explain a large part o the spatial and teporal characteristics o the odes involving the successive orders o branches relative to the irst ode deoring the trunk ( Fig. 8 ). The distribution o odal characteristics was particularly well predicted in the case o the tree with highest odal density and where the branch odes are the ost salient, i.e., the sypodial walnut tree. Moreover, in both trees, scaling laws were able to predict correctly the relative ranking o the dierent types o odes ( Fig. 8A ), validating the hypothesis that the diensional analysis o the syetric odes o idealized ractal trees can capture a large part o the scaling o odal settings in real trees (requencies and localization o bending and kinetic energy), although ore advanced analysis ay be conducted or onopodial trees. Such scaling law has two ajor uses. Fro a ethodological and practical point o view, the overall dynaics o a coplex tree can be reduced to (1) the easureent or estiation o the ost basal ode, which is the easiest to characterize and has been studied or odeled in nuerous studies (e.g., Gardiner, 199 ; Spatz and Zebrowski, 001 ); () a standard description o the branching ode, i.e., sypodial vs onopodial ode

13 Deceber 008] Rodriguez et al. Tree vibration odes 1535 ( artheley and Caraglio, 007 ); and (3) three siple bioetrical paraeters that have been easured in any bioechanical and ecological studies (e.g., McMahon and Kronauer, 1976 ). This copact description o the overall dynaics o a coplex tree is to be copared with the extensive work on detailed 3D digitizing ( Sinoquet and Rivet, 1997 ) ollowed by coplete odal analysis. It would be interesting though to test these scaling laws in trees o other species and other sizes, so that the accuracy in the prevision through these sipliied laws could be assessed ore copletely. Fro a ore undaental perspective, these scaling laws give direct insights into the signiicance o tree architecture and geoetry or its odal behavior and thus to its excitability to wind and its possible echanoperceptive control, as discussed next. Eects o architecture and bioetrical characteristics on odal content: Tuning and copartentalization oth area reduction ratios and the slenderness coeicient aect the relative requency and the location o odes (see Eqs. 16 and 1), whereas the branching angle only aects the spatial localization o the odes. In all the cases, the eects o the paraeters are nonlinear and ixed. For exaple, in the case o the sypodial tree, variations in λ and β both inluence the value o the requency o a given group o odes. In the natural ranges estiated ro our data ( Table 4 ), a decreasing λ (i.e., a higher reduction in the crosssectional area at branching) increases the relative requency o a given group o ode. A decreasing β (i.e., a tree with higher slenderness) also increases the relative requency o a given group o ode. It should be noted here that both λ and β have been reported to be under siilar control o wind echanoperception through thigoorphogenetic secondary growth responses ( Telewski, 006 ; Watt et al., 005 ). Thus, thigoorphogenetic responses ay be able to tune the ultiodal requencies range o the whole tree, whatever the genetic speciic traits o its architecture. It is, oreover, striking that two trees as geoetrically dierent as an old walnut tree and a young pine tree could present undaental odes in the range o Hz with a large nuber o their branch odes in the.5 3 Hz band, consistent with any reports in the literature (. Roan, Ecole Superieure de Physique et Chiie Industrielles de Paris (ESPCI), personal counication). This siilarity in odal requencies ay point toward soe odal tuning controlling the bioetrical paraeters o the trees (and thus o the scaling laws). The eectiveness o this accliation process reains to be studied, but the current study provides useul tools to do so. Last but not least, a very unexpected salient conclusion that is captured by the scaling laws is that branching and secondary growth are tuned so that the reduction o cross sectional area at branching points ( λ and eventually μ in onopods), induce a clear structural copartentalization o the odal spatial distribution and a scaling siilarity between successive odes. Whatever the architecture, odes have been ound to be ore and ore local as their odal requency increases. And both their odal requencies and odal ass are scaled recursively to that o the irst ode o the whole tree. These copartentalization and scaling siilarity are not echanical necessities. As previously stated, the elastic strain energy underlying odal behavior is distributed alost instantaneously over the whole structure; so that structural copartentalization and scaling siilarities result ro a speciic bioechanical design o the trees during their architectural developent. This speciic bioechanical design o the trees requires a consistent tuning o both (1) branching syetries within the architecture and () the secondary growth balance between parent and axillary branches (as reported by Watt et al., 005. Moreover, this consistent tuning should be eicient in highly dierent architectural patterns (onopods vs. sypods) and is thus very likely to have resulted ro adaptation. These structural copartentalization and scaling siilarities are probably iportant in aking the overall biological control over the ultiodal dynaics o the tree ore tractable, whatever its size. Assessing how this gain in control ay be beneicial or species adaptation and individual accliation to wind should be the atter o speciic uture investigations. Indeed, scaling laws give only approxiations, and clear dierences were ound in the odal spatial patterning between onopodial and sypodial trees. Moreover, trees in a orest stand ay have ore signiicant shoot abrasion or crown asyetry. At last, copetition or resources and photoorphogenetic responses to shade ay interact with the echanoperceptive accliation to wind ( Fournier et al., 005 ). ut soe eleents aecting the bioechanical signiicance o ultiodal scaling o trees to the response to wind load can already be directly discussed ro our results. Signiicance o ultiodal dynaics and scaling laws to the responses o trees to wind Wind excites trees through the drag orce applied to the constitutive eleents o the trees, branches, and possibly leaves or needles. Fro surace area considerations, ost o the drag thus occurs at in the distal, possibly leay, segents o the tree. All the odes in this study have a coon characteristic: their larger displaceents sits on the extreities o the tree. Thereore, they should reciprocally all be excited by a orce applied at the extreities, such as the wind-drag orce ( Gerardin and Rixen, 1994 ). Moreover, because wind spectra usually have a large requency band ( Raupach et al., 1996 ) overlying ost o the odal requencies o the considered odes, several odes ay be excited directly by highly luctuating winds. As a consequence, the two types o tree architectures studied here should have a dense ultiodal response to gusts involving a very signiicant contribution o branches o all the orders. Jaes et al. (006) and Spatz et al. (007) have argued that dynaics including branch deoration with close odal requencies could be beneicial to the tree by enhancing aerodynaic dissipation through a echanis called ultiple resonance daping or ultiple ass daping. A prerequisite or this echanis to occur is a ultiodal behavior o the tree, with high odal density in the requency range and signiicant branch deorations, exactly what was ound here or trees with contrasting architectures. This dense ultiodal dynaics, a consequence o the branched structure, can then be interpreted as a strategy to prevent the trunk ro bending excessively until the rupture. It should be noted, however, that the high odal density observed in our two trees did not reach the alost perect tuning in the odal requencies o branches larger than 0.5 reported by Spatz et al. (007) or a (onopodial) Pseudostuga enziesii tree. A siilar result in our study would have eant either λ and μ = 1 or β =, which can be rejected statistically in our two trees ( Table 4 ). However in our study, possible variations in the longitudinal Young s odulus along the branch (that have been reported by Spatz) were not considered. It would be interesting to urther investigate i the distribution