The Mechanical Design of Treesh the trunk and Branches of a tree grow they get not only longer

but also thicker. What law governs the relation between the twodimensions? The question is examined by hypothesis and experiment

]-"hc branching forms of trees aredeeply pleasing to us and wouldseem to serve the needs of the

n • themselves. In their proportions„,. is a harmony that makes us won-tr if we could discover a principle for

th.., mechanical design. Out of the inul-' natural principles that mightse diameter of a given limb or

could there be a single most Im*„ ,'ntone?

I -ually, of course, tliere is a singlem i important design principle both ina,! i e and in engineering. In choosinglli. .liametcr for the cables of a suspen-fJcHi bridge the engineer must maketii, in large enough to support tho load.J" •.truts of the same bridge are moro4 ult to design, because tho mecha-m i of failure may be cither bucklingoi aiding in compression, but a shortcl it with pencil and paper will settle•\ . h of the two presents the greatestll .it. and the bridge can be designedli mid tlie most obvious danger.

Then the bridgo is finished, it maytu . out to be considerably more cornel • altd as a system than it is as a mere• ■ i.iblage of parts. For example, windsni excite unforeseen vibrations; few»: have seen the motion pictures of theT .ma Narrows Bridgo breaking up in» hgjl wind will ever forget the specie i'\ After such a lesson everyone gen-•tally agrees that tho only good designb "lie that considers tho mechanical in-t.r.K tions of tho entire structure, notfit lhe stresses that might lie placed onc•!■ li part of it. Intuition tells us that theni' i lianical proportions of trees may bolu • il on just such an integrative prin-'i|> •. In what follows we shall explorei'« possibility in new theories and cx-|>u intents, but first let us review some<■' .hat is already known about howIn. - grow.

by Thomnu A. McMahon

and of plants in general, depends on theeffects of various plant hormones andinhibitors. Three major classes of hormones are now recognized: the auxins,the gibberellins and tho cytokinins. Thohormones that control Uio predominantfactor in tree growth, which is to saythe synthesis of new wood, are tho auxins. The most important auxin, bcta-indolylacctic acid (IAA), is synthesizedin apical plant tissues: tlie tips of shoots.

Long before the existence of auxinswas recognized their action on growingplants had been examined. Late in the19th century Charles Darwin and hisson observed the preferential bendingof grasses toward a light source. In thiscentury tho cause was determined to boa greater concentration of auxin in thotissue on tho less illuminated side of theplant stem. It was also found possiblero reproduce tho effect in total darkness.If the tip of a plant stem is removed

and auxin is supplied to one side of thestem only, cellular growth on tho sideof the stem deprived of auxin is slowed.Because the undeprived side continuesto grow normally the stem becomes bent.

Liko hormones in gencr '

of the plant that reactlug experiments in thv. .- -.Went of the University of Utrechtproved that auxin traveled in a preferred direction and ft— 'of travel was independity and diffusion. Worseedlings. Went first rerthe coleoptile, or first limoved an additional segment of theseedling stem. When hestem segment and replac— .. —Up on the stalk, the auxin would nottravel through the inverted sc)even though gravity favored move.......

• • " . - ; , ' : •■ " •■ • ' . • • > - * t - ^ T v ^ r - ! - ' ^ ' . •■ '

J S T R A H L E R O R D E R N U M B ! •

SYMMETRICAL PATTERN of ramification, characterised br tho Io,.rithmle -r.pl. {right). Sl.clin- .1 iho top. the ramificalioi, ,bifurcation of each iucce--ive branch (le/l), produce! ■ JeTel de- ti.t of four first-order ae-menH, Iho »econd-order -e-mem ," " " ' ' ' ' ' " : ' " - " -■ : • '■ : - - - -■ ■ • . „ , S t r . h l e r ' - • • . ( e m , d e v i l e d t o d e . i U

... .,.. applicable to any kind of branching p. in,

ASYMMETRICAL PATTERN of rar-'r- ......produco a level dei-endlng alope winwithin an order ii plotted on aFrom Ibe lop down tbe ramification- include aeven -e-menls of the

1 S T R A H L E R O R D E R N U M U i I

fir.-i order and one of the third order. Tbe icrond-order .n... n

in that direction. When he restored theinverted segments to their natural position and again replaced the tips, thetransport of auxin was resumed. Therate of transport, about 10 millimetersper hour, was too fast to l>c the result ofdiffusion alone.

Where trees are concerned one of thetarget tissues for auxin is the cambium,

activity that promotes the growth of newwoody tissue in the underlying xylemlayer.

The effect of auxins on cambial ncliv-'"' "'•*' also observed long before lhe

botanist Ludwig Jost found in numerousexperiments during tlie I8D0's that cambial activity was stimulated by growingbuds and young leaves. When theseparts of a tree were removed, the diameter of the tree did not increase. Afterthe auxins became known the study oftheir effects continued. For example, observations of many trees show that asspring advances, cambial activity is firstevident in the twigs. Tlie activation progresses slowly inward from the twigsalong the limbs and trunk for a periodof days until finally the tree is increasingin girth over the full length of the trunk.

In experiments where buds or twigsare removed from a tree the cambiumremains inactive and no new xylem isf . , r. „ . . , I . . . n i , , : „ . , jR „ , . . , „ , ! . . „

At points on the tree where buds •»"directly out of the bark the immcil .Ir!underlying cambium is found to I -J.live.

In addition to its seasonal role ">■'plays a part in the day-to-day I. e •'■trees. Consider a forest tree th.i h.-been shifted from itsition during u storm. Eventually ii »1probably grow straight again. A k:i iMgrowth known as reaction wood I nWalong one side or the other of thi d--placed stem and expands or conl'.u''powerfully enough to restore the ti"'1an upright position. Experiments in™cate the involvement of auxin 111 Wbending process. For example, "I"synthetic auxin is applied to test SfB*

■ ids'at tho site of application untilpine item bends away from tho

,ivth site. A similar unequal distribu-i„ of natural auxin has been measuredtree branches. The concentration of

„■ hormone is found to lie higher in(,,. Ijwer half of the limbs than'iu lhe^n-rhalf.

til of this suggests that in searching*» lor something that controls the in-i.^ing diameter of tree trunks andranches one is well advised-to seek a

m, ,!ijnism that is mediated by planthiiMiinnes. It is not difficult to imdcr-st.,11-1 the function the mechanism ful-"j!. ihe goal is tho production of a dura-bl tree. But what is it that signals them. I (or increased girth? And how is anin- . ..se recognized as being adequateHi: ihe growth signal then attenuated01 mmed off? It is at this point that ana-hi .1 models related to the forms and™.. .rtions of trees become pertinent.If .• are to discuss the strength of ait. lure and the loads the structurem liear, it is necessary to establishCC" .in facts about the structure itself.

r centuries observers of naturej. wondered if common rules could> und that would apply to such fun-i. 11 tally different kinds of ramifica-li . is branching in trees, in watersheddi age systems and in anatomicalil. lures such as blood vessels andhi n hial tubes. Over the past few dec-111 lhe ordering systems that were first1. loped by gcomorphologists to deli e drainago networks have been1. li used in quantitative studies ofr. thing patterns. One of tho mostk ly applied systems was devised byli '.eomorphologist Arthur N. Strahler.■irting at the point of ultimato rami-

. .111, Strahler assigns to these lowest-« 1 units, which wc can call links, theui eral I. In the description of a drain-IJj pattern these units are tho initial•> r streams; in the description of ali 'liey are the outermost twigs. When-"n 'wo such first-order links meet, they•' ine to form another link that con-in , onward. (In a drainage patternI* uitinuing link is in the direction of

.. as frequently happens in an

second-order link. The joining of twosecond-order links In turn produces athird-order continuation and so on untiltlio main stream, or the trunk of the tree.Is reached [see lower Illustration onopposite page].

Strahlcr'i system is further simplified,topologically speaking, by the introduc

tion of what can be called the segmentconcept For example, In the asymmetrical pattern in the lower illustration onthe opposito page, each single first-orderlink is also a segment So too aro anyadjoining links that belong to tho sameorder. Thus seven first-order links areillustrated, and each also ranks as a seg-

STRAHLER ORDER NUMBER

Wi Ihe continuation is assigned the■» 'al of tho higher order. Thus thopuiiitj- of either two first-order links orI iii.1 order link and a second-order oneP"»l.iecs a continuation in tho form of a A

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LENGTH (CENTIMETERS)

lion increases as the stiffness of i|lesprings is increased, and it decrease', uthe magnitude of tho masses is lurtased,It follows that when a tree is shaken iasuch a way that it sways from side to '.sit(that is, tlie lowest mode of vibrati ,,).the natural frequency of its to-anil domotion depends on the way both Mlness and mass aro distributed thrOu ;li-

. out the structure of the tree. -' | | 1 lie ax-head model of a tree limb , ...

diets that the natural frequency ol vibration for a branch bending up nd

|SJ L « down will bc inversely proportion;.! u,K ' . |hc square root of the length of l „

branch (given the value of beta as 3 J).This result is again independent of Ichoice of value for alpha. The ana' iiiis also valid for small-amplitude shal itgof a statically stable tapering col. ra-(such as the ax-hcad model standing jiright) and thus applies to whole tre. ...well as to lateral branches. Small I n

., ,.,., , ., , ,..., ,lflM should therefore have a higher nal ifrequency of vibration than large i aof the same species, and indivi ul

.„A. „n it,. dUirihuilon of branches should have a higher freq n•o- cy than whole trees. Tlie predicted a-

»•. poncnt in the frequency-length rel. imli'- is equal to beta minus 2. By way of. in-

parison, for geometric similarity I I-equals 1) the natural frequency is rcdieted to be inversely proportion, kilength and for static-stress simil ill(beta equals 2) the natural frequcn i.predicted to bc completely indepen mof length.

In order to test the prediction ol nmodel, my students and I last JIclamped a succession of tree branch Ba rigid bench vise, measured tho loi -lpath from the point of clamping t< **«terminal twig and then rhythmi. Inpushed on tltc branch to make it ! ■»up and down in the lowest mode ■• u>I b ra t i on . We reco rded the nu in l x ■»■cycles per unit of time and then clan • '

j the branch at another point and rep< tithe procedure.

We measured the natural frcqm *). not only of branches but also of the m.ihi

] " • steins of trees and even of whole t'"*i " with and without leaves. The ipecilin »we studied included poplar, larch, "j

"**• maple, red oak and white oak. Wcfn "'■that the relation between natural l»c*quency and length was close to ll«—— prediction of tho elastically similar .*•

4 0 0 8 0 0 8 0 0 , . 0 0 0 2 . 0 0 0 P ^ ^ ^ ^ ^ ^ ^ ^ - „ . , „

frequency-length relation was —.59., We found that among trees of II*pie. turned out lo have a natural frequency of

al,) lhan the .ame tree did when It wai in leaf «f species the natural frcquenc; "I.-.io tiPtn nf lhe tree was slic1''*

, identical length- This suggests that I

■ the main stem is mechanically sUH-We also found that the natural fre-„cy of leafless oaks and maples was

"ineen two and three times higherI., the natural frequency of these spe-• , when they are in leaf. Nonetheless,T rs in leaf fitted the model as well «Jle,,'i!.-ss ones did [see bottom illustrationou.rpositepage\.

;i,e vibration experiments proved tou sensitive test of the value assigned,, ,ta in tho ax-head model. As weV, seen, when the postulate of clasticsi, '.irity was incorporated in that mod-| • predicted that the frequency of vi-

w .,„ would be inversely proportionalu ..• square root of the branch length.I, not only tlie close agreement of thop. imental data with the model's prc-Jj' >ns that allows us to reject the twoJi natives to the elastic-similarity model omctric similarity and static-stressii, irity). Tlie first of the alternativesin redly predicts that the natural fro-q, cy will be inversely proportional totii ' ngth rather than to the square rooto! .• length. Tlie second predicts thecl. ly unrealistic condition that the nat-or frequency will l>c completely indo-p, ,nt of the length.ic primacy of the principle of elastic nilarity is implicit both in the com-n, m of natural tree morphology withth -cam, column and ax-head modelst is and in the studies of the relationl> ecn branch length, trunk length_ die natural frequency of vibration.A nr data support the conclusion thattl ..rinciplo underlying the mechanical,: -ii of most if not all trees is tbe main-tr ice of elastic similarity. This leadslo further conclusion. Something fun-d., -iital about tho process of growthp. nls a tree limb from being deflect-et! / its own weight past some criticalr>. When tho curvature induced by*. • ...ceds a specific threshold val-u: mutt be true that the limb aiitorn., .illy begins to increase in diameter.

I ■> conventional wisdom among sil-•iv urists that trees raised in a green-h... will show greater growth in diam-ctn .1 given a shaking for a brief intervalts. I .lay; trees grown under identicalCon linns but without such stimulationdu i t grow as thick. By the same tokentii, that aro grown outdoors must notbe - ipported by guy lines for too long;oth., vije they will have such tall and«-ni vr proportions that tliey will buckle**'",. the guys are removed. All this in-j ■"million, both quantitative and anec-

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