Unifying constructal theory of tree roots, canopies and forests

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Unifying constructal theory of tree roots, canopies and forests

A. Bejan a,!, S. Lorente b, J. Lee a

a Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USAb Laboratoire Materiaux et Durabilite des Constructions (LMDC), Universite de Toulouse, UPS, INSA, 135 Avenue de Rangueil, F-31077 Toulouse Cedex 04, France

a r t i c l e i n f o

Article history:Received 30 October 2007Received in revised form14 June 2008Accepted 27 June 2008Available online 2 July 2008

Keywords:Constructal theoryDesign in natureRootsTreesForestsLeonardo’s ruleFibonacci sequenceZipf distributionEiffel Tower

a b s t r a c t

Here, we show that the most basic features of tree and forest architecture can be put on a unifyingtheoretical basis, which is provided by the constructal law. Key is the integrative approach tounderstanding the emergence of ‘‘designedness’’ in nature. Trees and forests are viewed as integralcomponents (along with dendritic river basins, aerodynamic raindrops, and atmospheric and oceaniccirculation) of the much greater global architecture that facilitates the cyclical flow of water in nature(Fig. 1) and the flow of stresses between wind and ground. Theoretical features derived in this paper are:the tapered shape of the root and longitudinally uniform diameter and density of internal flow tubes,the near-conical shape of tree trunks and branches, the proportionality between tree length and woodmass raised to 1/3, the proportionality between total water mass flow rate and tree length, theproportionality between the tree flow conductance and the tree length scale raised to a power between1 and 2, the existence of forest floor plans that maximize ground-air flow access, the proportionalitybetween the length scale of the tree and its rank raised to a power between !1 and !1/2, and theinverse proportionality between the tree size and number of trees of the same size. This paper furthershows that there exists an optimal ratio of leaf volume divided by total tree volume, trees of the samesize must have a larger wood volume fraction in windy climates, and larger trees must pack more woodper unit of tree volume than smaller trees. Comparisons with empirical correlations and formulas basedon ad hoc models are provided. This theory predicts classical notions such as Leonardo’s rule, Huber’srule, Zipf’s distribution, and the Fibonacci sequence. The difference between modeling (description) andtheory (prediction) is brought into evidence.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Trees are flow architectures that emerge during a complexevolutionary process. The generation of the tree architecture isdriven by many competing demands. The tree must catchsunlight, absorb CO2 and put water into the atmosphere, whilecompeting for all these flows with its neighbors. The tree mustsurvive droughts and resist pests. It must adapt, morph and growtoward the open space. The tree must be self-healing, to survivestrong winds, ice accumulation on branches and animal damage.It must have the ability to bulk up in places where stresses arehigher. It must be able to distribute its stresses as uniformly aspossible, so that all its fibers work hard toward the continuedsurvival of the mechanical structure.

On the background of this complexity in demands andfunctionality, two demands stand out. The tree must facilitatethe flow of water, and must be strong mechanically. The demandto pass water is made abundantly clear by the strong geographical

correlation between the density (and sizes) of trees and the rate ofrainfall (Fig. 1). It is also made clear by the dendritic architecture,

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Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

Fig. 1. The physics phenomenon of generation of flow configuration facilitates thecircuit executed by water on the globe. Examples of such flow configurations areaerodynamic droplets, tree-shaped river basins and deltas, vegetation, and allforms of animal mass flow (running, flying, swimming).

0022-5193/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2008.06.026

! Corresponding author. Tel.: +1919 660 5314; fax: +1919 660 8963.E-mail address: [email protected] (A. Bejan).

Journal of Theoretical Biology 254 (2008) 529–540


which is the best way to provide flow access between one pointand a finite-size volume (Bejan, 1997). The demand to be strongmechanically is made clear by features such as the tapered trunksand limbs with round cross-section, and other design-like featuresidentified in this article. These features of designedness in solidstructures facilitate the flow of stresses, which is synonymous withmechanical strength.

According to constructal theory, ‘‘plants (vegetation) occur andsurvive in order to facilitate ground-air mass transfer’’ (Bejan,2006, p. 770). Recently, constructal theory (Bejan, 1997, 2000) hasshown that dendritic crystals such as snowflakes are the mosteffective heat-flow configurations for achieving rapid solidifica-tion (Bejan, 1997; Ciobanas et al., 2006). The same mental viewingwas used to explain the variations in the morphology of stonycorals and bacterial colonies and the design of plant roots (Miguel,2006; Biondini, 2008). The 23-level architecture of the lung (Reiset al., 2004), the scaling laws of all river basins (Reis, 2006; Bejan,2006), and the macroscopic features (speeds, frequencies, forces)of all modes of animal locomotion (flying, running, swimming)(Bejan and Marden, 2006) were attributed to the same evolu-tionary principle of configuration generation for greater flowaccess in time (the constructal law).

In summary, there is a renewed interest in explaining the‘‘designedness’’ of nature based on universal theoretical principles(Turner, 2007), and constructal theory is showing how to predictthe generation of natural configuration across the board, frombiology to geophysics and social dynamics (for reviews, see Bejanand Lorente, 2006; Bejan, 2006; Bejan and Merkx, 2007).

In this paper, we rely on constructal theory in order to constructbased on a single principle the main features of plants, from root andcanopy to forest. We take an integrative approach to trees as live flowsystems that evolve as components of the larger whole (theenvironment). We regard the plant as a physical flow architecturethat evolves to meet two objectives: maximum mechanical strengthagainst the wind, and maximum access for the water flowingthrough the plant, from the ground to the atmosphere.

Ours is a physics paper rooted in engineering. The purpose ofour work is to demonstrate that the existence of tree-likearchitecture can be anticipated as a mental viewing based on theconstructal law. The work is purely theoretical. Althoughcomparisons with natural forms are made, the work is notintended to describe and correlate empirically the diversity ofplant measurements found in nature. Although we are not nearlyas familiar as our biology colleagues with the sequence oftheoretical and empirical advances made on vegetation morphol-ogy, in constructal theory we have a physics method with whichwe have predicted natural flow design across the board (Bejan andLorente, 2006). We bring to this table of discussion the tools ofstrength of materials, fluid mechanics, and, above all, theengineering thinking of multi-objective design. We believe thatour physics work will be of interest because of its engineeringorigins and purely theoretical character and message.

2. Root shape

The plant root is a conduit shaped in such a way that itprovides maximum access for the ground water to escape aboveground, into the trunk of the plant. The ground water enters theroot through all the points of its surface. In the simplest possibledescription, the root is a porous solid structure shaped as a bodyof revolution (Fig. 2). The shape of the body [L, D(z)] is not known,but the volume is fixed:

V "Z L

0

p4

D2 dz (1)

The flow of water through the root body is in the Darcy regime.The permeability of the porous structure in the longitudinaldirection (Kz) is greater than the permeability in the transversaldirection (Kr). Anisotropy is due to the fact that the woodyvascular tissue (the xylem) is characterized by vessels and fibersthat are oriented longitudinally.

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Nomenclature

a, b factors, Eqs. (8), (9), (23), (28) and (29)a0 factor, Eq. (25)A area (m2)AB branch cross-section at the trunk (m2)AL leaf area distal to stem (m2)At tree cross-section at x (m2)AW sapwood cross-section (m2)c1, c2 factors, Eqs. (48) and (49)C global flow conductance, Eq. (50)CD drag coefficientD diameter (m)Dc canopy diameter (m)Dc,B diameter of branch canopy (m)DL diameter at z " L (m)Dt trunk diameter (m)Dt,B diameter of branch (m)F0 drag force per unit length (N/m)h frustum height (m)HV Huber valueIt area moment of inertia (m4)kr radial specific conductivity m/(s Pa)ks stem specific conductivity m/(s Pa)Kr radial permeability (m2/s)Kx, Kz longitudinal permeability (m2/s)L length (m)

LB branch length (m)LSC leaf specific conductivitym, n exponents, Eqs. (8), (9) and (23)m bending moment (N m)_m mass flow rate (kg/s)_mB branch mass flow rate (kg/s)

p exponentP pressure (Pa)Pg ground pressure (Pa)PL pressure at z " L (Pa)Pv vapor pressure (Pa)P0 branch tip pressure (Pa)Ri rank of trees of size Di

sm maximum bending stress (N/m2)u Darcy (volume averaged) longitudinal velocity (m/s)uB branch Darcy longitudinal velocity (m/s)v Darcy radial velocity (m/s)V wind speed (m/s)V volume (m3)VT total volume (m3)w wood volume fractionx, z longitudinal coordinates (m)Xs side of square (m), Fig. 7bXt side of equilateral triangle (m), Fig. 6bm viscosity (kg/s m)n kinematic viscosity (m2/s)r density (kg/m3)

A. Bejan et al. / Journal of Theoretical Biology 254 (2008) 529–540530


We assume that the (L, D) body is sufficiently slender, so thatthe pressure inside the body depends mainly on longitudinalposition, P#r; z$ % P#z$. This slenderness assumption is analogousto the slender boundary layer assumption in boundary layertheory. For Darcy flow, the z volume averaged longitudinalvelocity is given by

u " !Kz

mdPdz

(2)

where m is the fluid viscosity. Because of the P#r; z$ % P#z$assumption, for the transversal volume averaged velocity v(oriented toward negative r) we write approximately:

v%Kr

mPg ! P#z$

D=2(3)

The definition of the radial permeability (Kr) of the root bodyas a Darcy porous medium is Eq. (3). This definition is consistentwith Eq. (2), which is the definition of the longitudinalpermeability of the root as a nonisotropic Darcy porousmedium (e.g., Nield and Bejan, 2006). The directional permeabil-ities Kz and Kr are two constants. The radial permeability Kr

should not be confused with the concept of radial waterconductivity kr, which is defined as the ratio between the radialflux of water and the radial pressure difference [e.g., Eq. (3.3) inRoose and Fowler, 2004].

The ground-water pressure (Pg) outside the body is assumedconstant. This means that in this model the hydrostatic pressurevariation with depth Pg(z) is assumed to be negligible, and that theroot sketched in Fig. 2 can have any orientation relative to gravity.Ground level is indicated by z " L: here the pressure is PL, and islower than Pg. Throughout the body, P(z) is lower than Pg, and theradial velocity v is oriented toward the body centerline.

The conservation of water flow in the body requires

d _m " rpDv dz (4)

where _m is the longitudinal mass flow rate at level z:

_m " rp4

D2u (5)

and r is the density of water. Eqs. (4) and (5) yield

ddz#D2u$ " 4vD (6)

Summing up, the three Eqs. (2), (3) and (6) should be sufficientfor determining u(z), v(z) and D(z) when the length L is specified.Here, the challenge is of a different sort (much greater). We mustdetermine the shape [L, D(z)] that allows the global pressuredifference (Pg!PL) to pump the largest flow rate of water to theground level:

_mL " rp4

D2#L$u#L$ (7)

subject to the volume constraint (1). Instead of trying a numericalapproach or one based on variational calculus, here we use a muchsimpler method. We assume that the unknown function D(z)belongs to the family of power-law functions:

D " bzm (8)

where b and m are two constants. We also make the assumptionthat the function P(z) belongs to the family represented by

Pg ! P#z$m=Kz

" azn (9)

where a and n are two additional constants. When we substituteassumptions (8) and (9) into Eqs. (2) and (3), and then substitutethe resulting u and v expressions into Eq. (6), we obtaintwo compatibility conditions for the assumptions made inEqs. (8) and (9):

m " 1 (10)

b2n#n& 1$ " 8Kr

Kz(11)

The volume constraint (1) yields a third condition:

b2L3 "12p V (12)

A fourth condition follows from the statement that the overallpressure difference is fixed, which in view of Eq. (9) means that

Pg ! PL

m=Kz" aLn; constant (13)

Finally, the mass flow rate through the z " L end of the body is, cf.Eq. (7):

_mL " rp4#bL$2

Kz

md#Pg ! P$

dx

! "

z"L

" rp4

b2anLn&1 (14)

for which b(n) and L(n) are furnished by Eqs. (11) and (12). Theresulting ground-level flow rate is

_mL " rp4#aLn$ 8

Kr

Kz

! "2=3 12p V

! "1=3 n1=3

#n& 1$2=3(15)

with the observation that (aLn) is a constant, cf. Eq. (13).In conclusion, _mL depends on root shape (n) according to the

function n1/3/(n+1)2/3. This function is maximum when

n " 1 (16)

Working back, we find that the constructal root design must havethis length and aspect ratio:

L "3VKz

pKr

! "1=3

(17)

LDL"

12

Kz

Kr

! "1=2

(18)

The constructal root shape is conical. The slenderness of this coneis dictated by the anisotropy of the porous structure (Kz/Kr)

1/2.

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Fig. 2. (a) Root shape with power-law diameter; (b) constructal root design:conical shape and longitudinal tubes with constant (z-independent) diameters,density, u and v.

A. Bejan et al. / Journal of Theoretical Biology 254 (2008) 529–540 531


The root is more slender when the vascular structure is morepermeable longitudinally.

Another important feature of the discovered root geometry isthat the longitudinal volume averaged fluid velocity (u) isindependent of longitudinal position (z), because n " 1 meansthat dP/dz " constant, and

u "Kz

mPg ! PL

L#0ozoL$ (19)

_mL " rp4#bL$2u "

Kz

n #Pg ! PL$#3V$1=3 pKr

Kz

! "2=3

(20)

The morphological implications of this theoretical feature areimportant. If the porous structure is a bundle of tiny capillarychannels, then the fluid velocity through each tube must beconstant, and must not depend on the size of the root cross-section (p/4)D2(z) that the channel pierces. On the other hand,earlier work on constructal design (Bejan and Lorente, 2004) hasshown the following: flow strangulation is not good for flowperformance, the constructal configuration of a long capillarywith specified flow rate and volume is the one where the cross-section does not vary with longitudinal position, and the cross-section is round. Combining this with the new conclusion that umust not depend on z, we discover the internal structure of theconstructal root body. The longitudinal tubes must be round, withdiameters that do not vary with z, even though some tubes arelonger than other tubes. The external shape and internal structureof the root body discovered in this section are sketched on theright side of Fig. 2.

Another feature of the constructal root design is visible inEq. (3). Because both (Pg!P) and D are proportional to z, weconclude that v must also be z-independent. One can show that

vu"

Kr

Kz

! "1=2

(21)

The anisotropy of the vascular porous structure dictates the ratiobetween constant-v and constant-u, in the same way that itdictates the root slenderness ratio DL/L, cf. Eq. (18).

There is a considerable body of literature on the modeling ofwater flow through roots, and a common assumption is that theroot is a porous conduit with constant diameter. The analysis thenyields a pressure that varies nonlinearly (exponentially) along theconduit (Landsbert and Fowkes, 1978; Frensch and Stendle, 1989;Roose and Fowler, 2004). This is consistent with the analysisshown in this section, for if this analysis is repeated by postulatingthat D is constant, then in place of the present conclusion (dP/dz " constant) we find that P(z) must vary exponentially. How-ever, measurements made on root segments show that thepressure does not vary linearly (Zwieniecki et al., 2003; McElroneet al., 2004). This is a good opportunity for next-generationanalytical and experimental work. For example, the Landsbert andFowkes (1978) modeling should be combined with the optimaltapering of the conduits, i.e., with the constructal law offacilitating flow access by allowing the flow geometry to morph.

3. Trunk and canopy shape

The water stream guided by the root from underground toground level continues to flow upward through the trunk andcanopy of the plant. To continue with the same analytical ease asin the analysis of root geometry, for the trunk and canopy of theplant we make the simplifying assumptions hinted at in Fig. 3,which is based on a problem proposed in Bejan (2006,pp. 831–832). We assume that both the canopy and the trunkare sufficiently slender. This allows us to analyze the forces exerted

by the wind on the canopy as a problem of two-dimensional flow,in a horizontal plane that cuts the trunk and the canopy.

The trunk and the canopy are modeled as two bodies ofrevolution, with unknown diameters Dt(x) and Dc(x), where x ismeasured downward from the top of the tree. The drag force perunit length (x) experienced by the tree canopy is

F 0 " CDDc ' 12rV2 (22)

where V is the horizontal wind speed and Dc(x) is the radius of thecanopy at the distance x from the tree top. We assume that theReynolds number VDc/n is greater than 103, so that the dragcoefficient CD is a constant approximately equal to 1.

To give our search for geometry sufficient generality, assumethat the canopy has a shape that belongs in the family of power-law functions:

Dc " axn (23)

where a is a constant and the shape exponent n is not known. Thebending moment experienced by the trunk at the distance x fromthe tree top is

M#x$ "Z x

0F 0#x! x$dx "

a0xn&2

#n& 1$#n& 2$(24)

where a0 is another constant:

a0 "a2rV2CD (25)

We now turn our attention to the maximum bending stresses inthe cross-section of the trunk of diameter Dt(x):

sm "M#x$It#x$

Dt#x$2

(26)

where It " pD4t =64. The stress sm occurs in the dorsal and ventral

fibers of the trunk, as the trunk bends in the wind that pushes thecanopy. ‘‘Optimal distribution of imperfection’’ (Bejan, 2000)means that sm must be the same over the entire height of thetree. According to Eq. (24), the trunk diameter must vary as

Dt#x$ "32a0=p

sm#n& 1$#n& 2$

# $1=3

x#n&2$=3 (27)

This is an important result, but it is not the end of the story. Itsays that if we know the canopy shape (n), then we can predict thetrunk shape, and vice versa (Fig. 4). To determine the trunk andcanopy shapes uniquely, we need an additional idea (Section 4).

If the canopy is shaped as a cone (n " 1), then the trunk is alsoshaped as a cone, Dt/x " constant. Fig. 4 shows that if the canopyhas a round top (e.g., n " 1/2) then the trunk diameter must varyas Dt/x

5/6 " constant, which is not much different than Dt/x " constant. If the canopy has a very sharp tip (e.g., n " 3/2),then Dt(x) must vary as Dt/x

7/6 " constant, which again is not far

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Fig. 3. Slender tree canopy and trunk exposed to a horizontal wind with uniformvelocity.



off the conical trunk shape. In sum, we have discovered that theshape of the trunk that is uniformly stressed is relativelyinsensitive to how the canopy is shaped. A conical trunk isessentially a uniform-stress body in bending for a wide variety ofcanopy shapes that deviate (concave vs. convex) from the conicalcanopy shape sketched in Fig. 3.

A simpler version of the problem solved in this section is tosearch for the optimal shape of the trunk Dt(x) when thereis no canopy. The trunk alone is the obstacle in the wind,and its bending is due to the distributed drag force F0 ofEq. (22), in which Dc is replaced by Dt. The analysis leads toEq. (27) where M(x) varies as xn+2, and sm (constant) isproportional to M#x$=D3

t . The conclusion is that the trunk (orsolitary pole) is the strongest to bending when it is conical, n " 1.The same result follows from the subsequent discussion ofEq. (27), if we assume Dc " Dt.

A famous structure that only now reveals its bending-resistance design is the Eiffel Tower (Science et Vie, 2005). Theshape of the structure is not conical (Fig. 4) because in addition tobending in the wind, the structure must be strong in compression.The optimization of tower shape for uniform distribution ofcompressive stress leads to a tower profile that becomesexponentially narrower with altitude. The shape of a tower thatis uniformly resistant to lateral bending and axial compression isbetween the conical and the exponential. This apparent ‘‘im-perfection’’ (deviation from the exponential) of the Eiffel Towerhas been a puzzle until now (see the end of Section 4).

This discussion of the Eiffel Tower also sheds light on a majormechanical difference between the present theory and the modelof West et al. (1999). In the present work, the mechanical functionis to resist bending due to horizontal wind drag, as in the uppersection of the Eiffel Tower. In the model of West et al., themechanical function is to resist buckling under its own weight, onthe vertical. Of course, all modes of resisting fracture areimportant, but, which is the more important? Buckling is not,because the weight of the tree is static, totally independent of thenotoriously random and damaging behavior of the flowingenvironment. The wind is much more dangerous. Record breakingwind speeds make news all over the globe, and their combinedeffect can only be one: the cutting of the trunks, branches andleaves to size. What is too long or sticks out too much is shavedoff. The tree architecture that strikes us as pattern today (i.e., the

emergence of scaling laws) is the result of this never-endingassault.

4. Conical trunks, branches and canopies

The preceding section unveiled the architecture of a tree thathas evolved, so that its stresses flow best and its maximumallowable stress is distributed uniformly. This tree supports thelargest load (i.e., it resists the strongest wind) when the treevolume is specified. Conversely, the same architecture withstandsa specified load (wind) by using minimum tree volume. Insummary, the multitude of near-conical designs discovered inEq. (27) and Fig. 4 refer to the mechanical design of the structure,i.e., to the flow of stresses, not to the flow of fluid that seeps fromthick to thin, along the trunk and its branches.

There is no question that the maximization of access for fluidflow plays a major role in the configuring of the tree. This is whythe tree is ‘‘tree-shaped’’, dendritic, one trunk with branches, andbranches with many more smaller branches. How do the designsof Eq. (27) facilitate the maximization of access for fluid flow?

The answer is provided by the constructal root discovered inSection 2 and Eqs. (17)–(21). The constructal shape for a bodypermeated by Darcy flow with two permeabilities (Kz, Kr) isconical. The longitudinal and lateral seepage velocities (u, v) areuniform, independent of the longitudinal position z. For a root, thelateral seepage is provided by direct (contact) diffusion from thesoil, and indirect seepage from root branches, rootlets and roothairs. For the tree trunk above the ground, the lateral flow thataccounts for v is facilitated (ducted) almost entirely by lateralbranches. Above the ground, the lateral v is concentrated discretelyin branches that are distributed appropriately along and aroundthe trunk (see the discussion of the Fibonacci sequence at the endof this section).

The theoretical step that we make here is this: the constructalflow design of the root is the same as the flow design of the trunkand canopy. From this we deduce that out of the multitude ofnear-conical trunk shapes for wind resistance, Eq. (27), theconstructal law selects the conical shape, n " 1. The conical shapeis also the constructal choice for the large and progressivelysmaller lateral branches, provided that their mechanical design isdominated by wind resistance considerations, not by the

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Fig. 4. Three canopy shapes showing that the optimal trunk shape is near-conical in all cases.



resistance to their own body weight. We return to this observationin the last paragraph of this section.

Recognition of the conical trunk and canopy shapes means thatthe analysis in this section begins with Eqs. (18) and (23), whichfor the tree trunk and canopy reduce to

Dt#x$x" 2

Kr

Kx

! "1=2

" b (28)

Dc#x$x" a (29)

Here, it should be noted that for the tree trunk the axialcoordinate (x) is measured downward (from the tree top, Fig. 3),whereas the axial coordinate of the root (z) is measured upward(from the root tip, Fig. 2). The proportionality between Dt(x) andDc(x) is provided by Eq. (27) with n " 1, in combination with Eqs.(25), (28) and (29):

Dc#x$Dt#x$

"ab"

3psm

2CDrV2

Kr

Kx(30)

Eq. (30) recommends a large Dc/Dt ratio for trees with hard woodin moderate winds, and a small Dc/Dt ratio for trees with softwood in windy climates. A hard-wood example is the walnut tree(Juglans regia) with sm ’ 1:2' 108 N/m2, in a mild climaterepresented by V(50 km/h (14 m/s). Eq. (30) with CD(1 yieldsDc/Dt(2.42'106(Kr/Kx)walnut and, after additional algebra,Dc=Ltrunk(4:8' 106#Kr=Kx$

3=2walknut . The corresponding estimates

for a pine tree (Pinus silvestris) with sm ’ 6:6' 107 N/m2 in awindy climate with V(100 km/h (28 m/s) are Dc/Dt(3.4'105(Kr/Kx)pine and Dc=Ltrunk(6:8' 105#Kr=Kx$

3=2pine.

How many branches should be placed in the canopy, and atwhat level x? We answer this question with reference to Fig. 5,where the aspect ratios of the trunk (Dt/x " b) and canopy (Dc/x " a) also hold for the branch LB(x) located at level x:

Dt;B

LB" b;

Dc;B

LB" a (31)

Furthermore, in accordance with Eq. (29) for the canopy, Dc(x) isthe same as 2LB(x), which means that

LB#x$ " 12ax (32)

Dt;B#x$ " 12abx (33)

Dc;B#x$ " 12a2x (34)

A single branch LB(x) resides in a frustum of the conicalcanopy: the frustum height is h(x) and the base radius is LB(x).In the center of this frustum, there is a trunk segment (anotherconical frustum) of height h(x) and diameter Dt(x). The trunkfrustum can be approximated as a cylinder of diameter Dt(x).The total flow rate of fluid that flows laterally from this trunksegment is

_mB " rvpDth (35)

If uB is the longitudinal fluid velocity along the branch LB, then thesame fluid mass flow rate can be written as

_mB " ruBp4

D2t;B (36)

where Dt,B is the diameter of branch LB at the junction with thetrunk. Eliminating _mB between Eqs. (35) and (36), and using Eqs.(28), (32) and (33), we find that h is proportional to x:

hx"

uB

ua2

8(37)

The ratio uB/u is a constant determined as follows. Let P(x) bethe pressure at level x inside the trunk, and P0 the pressure at the

tip of the trunk (x " 0). The pressure at the tip of the branch LB isalso P0. In accordance with Eq. (19), we write

u "Kx

mP#x$ ! P0

x(38)

uB "Kx;B

mP#x$ ! P0

LB(39)

which yield

uuB"

Kx

Kx;B

LB

x(40)

It is reasonable to assume that the longitudinal permeability ofthe wood to be the same in the trunk and the branch, Kx % Kx;B,such that Eq. (37) reduces to

h " 14ax " 1

4LB (41)

In conclusion, the vertical segment of trunk (h) that isresponsible for the flow rate into one lateral branch is propor-tional to the length of the branch. Another dimension that isproportional to LB(x) is the diameter of the conical ‘‘branchcanopy’’ circumscribed to the horizontal LB, namely Dc,B " aLB, cf.Eq. (34). Comparing h with Dc,B, we find that

h#x$Dc;B#x$

"1

2a(42)

which is a constant of order 1. In other words, there is room in theglobal canopy (L, Dc) to install one LB-long branch on every h-tallsegment of tree trunk. The geometrical features discovered in thissection have been sketched in Fig. 5.

One of the reviewers of the original manuscript asked us tocompare this tree architecture with that of the model of West etal. (1999). This was a great suggestion because it leads to animportant theoretical discovery that is hidden in the mass-conservation analysis that led to Eq. (41). The discovery is thatLeonardo’s rule (e.g., Horn, 2000; Shinozaki et al., 1964) isdeducible from Eq. (41), in these steps. The trunk cross-sectionalarea at the distance x from the tip is At#x$ " #p=4$b2x2. At the topof the h frustum, it is At#x! h$ " #p=4$b2#x! h$2. The reduction intrunk cross-sectional area from x to x!h is DAt " At#x$ ! At#x! h$.The cross-sectional area of the thick end of the single branchallocated to h is AB " #p=4$D2

t;B " #p=4$b2L2B. The ratio between the

decrease in trunk cross-sectional area and the branch cross-sectional area allocated to that decrease is, after some algebra,DAt=AB " #2=a$)1! #a=8$*. In view of Eq. (42), where #1=2a$(1,according to constructal theory the ratio DAt=AB must be aconstant of order 1.

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Fig. 5. Conical canopy with conical branches and branch-canopies.



The area ratio would have been exactly 1 according toLeonardo’s rule, which was based on visual study and drawingsof trees. This rule is predicted here based on the constructal lawand other first principles such as the conservation of water massflow rate. In West et al.’s (1999) model, this rule was assumed, notpredicted. It was assumed along with several other assumptions(e.g., the tree-shaped structure), so that the model could becomecompact and useful as a facsimile—as a description—of the realtree, just like Leonardo’s observations. It is because of suchassumptions that the allometric relations derived algebraicallyfrom West et al.’s (1999) model are ‘‘description’’, not prediction.This remark is necessary because it contradicts West et al.’s use ofthe words ‘‘predicted values’’ in the reporting of their derivations(e.g., Table 1, p. 667). Additional comments on West et al.’s modelare provided by Kozlowski and Konarzewski (2004) and Makelaand Valentine (2006).

In the present paper, the tree architecture and the tapering ofits limbs are deduced from a single postulate which is theconstructal law. Furthermore, because there is one lateral branchper trunk segment h(x), and because h decreases in proportionwith x, the best way to fill the tree canopy with the canopies of thelateral branches is by arranging the branches radially, so that theyfill the ‘‘alveoli’’ created in the canopy ‘‘cone’’ by two counter-rotating spirals that spin around toward the top of the tree canopy.When one counts the sequence in which these alveoli arrangethemselves up the trunk, one discovers the Fibonacci sequence(e.g., Livio, 2002).

Like Leonardo’s rule, the Fibonacci sequence is the result ofEq. (42), the predicted conical canopy shape, and the geometricrequirement that the next branch and canopy should shoot laterallyinto the space that is impeded the least by the branch canopiessituated immediately above and below. The need of minimuminterference between branches is a restatement of the constructallaw, i.e., the tendency to morph to have greater flow access for waterfrom ground to wind. Each branch reaches for the pocket of volumethat contains the least humid air flow. This principle is universal, andis fundamentally different than ad hoc statements such as ‘‘stemsgrow in positions that would optimize their exposure to sun, rain,and air’’ (Livio, 2002), and ‘‘phyllotaxis simply represents a state ofminimal energy for a system of mutually repelling buds’’ (Livio,2002; after Douady and Couder, 1992).

The tree structure discovered step by step up to this pointconsists of cones inside cones. The large conical trunk and canopyhosts a close packing of smaller conical branches and conical branchcanopies. One can take this construction further to smaller scales,and see the architecture of each branch as a conical canopy packedwith smaller conical branches and their smaller canopies. In such aconstruction, the wood volume is a fraction of its total volume, i.e., afraction of the volume of the large canopy, which scales as L3. Fromthis follows the prediction that the trunk length L must beproportional to the total wood mass raised to the power 1/3. Thisprediction agrees very well with measurements of five orders ofmagnitude of tree mass scales (e.g., Table 2 in Bertram, 1989).

In closing, we return to the Eiffel Tower discussed at the end ofthe preceding section, where we noted that strength in compres-sion (under the weight) near the base was combined withstrength in bending (subject to lateral wind) in the upper bodyof the tower. This discussion is relevant in the modeling of thehorizontal branch, which in this section was based on theassumption that the loading is due to lateral wind. The branchis also loaded in the vertical direction, under its own weight. If weassume that the distributed weight of the branch is the only load,then the branch shape of constant strength (i.e., with x-independent sm) has the form D " ax2, where a is constant. Sucha branch has zero thickness in the vicinity of the tip (dD/dx " 0 atx " 0+), and is not a shape found in nature. This result alone

indicates that the tips of branches are not shaped by weightloading alone, and that wind loading (which prescribes D " axand finite D at small x) is the more appropriate model there. Forthe thick end of the branch, it can be argued that D " ax2 is arealistic shape, and that near the trunk the weight loading of thebeam is the dominant shaping mechanism, just like in the EiffelTower near the ground.

5. Forest

Forests are highly complex systems, and their study hasgenerated a significant body of literature (for reviews, see Keittet al., 1997; Urban et al., 1987). Multi-scale models of landscapepattern and process are being applied, for example, models withspatially embedded patch-scale processes (Weishampel andUrban, 1996). To review this activity is beyond our ability, and isnot our objective. Here we continue on the constructal path tracedup to this point (Fig. 1): if the root, trunk and canopy architectureis driven by the tendency to generate flow access for water, fromground to air, then, according to the same mental viewing (i.e.,according to the same theory), the forest too should have anarchitecture that promotes flow access.

The fluid flow rate ducted by the entire tree from the ground tothe tips of the trunk and branches is:

_m " rup4

D2t #x " L$ "

p4

b2

anKx)P#x " L$ ! P0*Dc#x " L$ (43)

where x " L indicates ground level and Dc(x " L) is the diameter ofthe canopy projected as a disc on the ground. The importantfeature of the tree design discovered so far is the proportionalitybetween _m and Dc(x " L). This also means that the total mass flowrate is proportional to the tree height L. This proportionality willbe modified somewhat when we take into account the additionalflow resistance encountered by _m as it flows from the smallestbranches (P0) through the leaves and into the atmosphere (Pa). SeeSection 6.

Seen from above, an area covered with trees of many sizes (Dc,i)is an area covered with fluid mass sources ( _mi), where each _mi isproportional to the diameter of the circular area allocated to it.From the constructal law of generating ground-to-air fluid flowaccess follows the design of the forest.

The principle is to morph the area into a configuration with masssources (or disc-shaped canopy projections) such that the total fluidflow rate lifted from the area is the largest. From this invocation ofthe constructal law follows, first, the prediction that the forest musthave trees of many sizes, few large trees interspaced with more andmore numerous smaller trees. This is illustrated in Fig. 6a with atriangular area covered by canopy projections arranged according tothe algorithm that a single disc is inserted in the curvilinear trianglethat emerges where three discs touch. If the side of the large triangleis Xt, then the diameter of the largest canopy disc is D0 " Xt, and thenumber of D0-size canopies present on one Xt triangle is n0 " 1/2.For the next smaller canopy, the parameters are D1 " (3!1/2!1/2)Xt

and n1 " 1. At the next smaller size, the number of canopies isn2 " 3, and the disc size is D2 " 0.0613Xt. The constructioncontinues in an infinite number of steps (n3 " 3, n4 " 6, y) untilthe Xt triangle is covered completely. The image that would resultfrom this infinite compounding of detail would be a fractal. The totalfluid flow rate vehicled by the design from the triangular area of Fig.6a is proportional to

ma "X1

i"0

niDi "12

D0 & D1 & 3D2 & + + + " #0:761& + + +$Xt (44)

Because a canopy disc D contributes more to the global production(m) when D is large and when the number of D-size discs is large,

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a better forest architecture is the one where the larger discs are morenumerous. This observation leads to Fig. 6b, where the Xt triangleis covered more uniformly by larger discs, in this sequence: D0 "[(3!1/2+1)/2]Xt and n0 " 1/2, D1 " [(31/2!1)/2]Xt and n1 " 1, D2 "[(1!3!1/2)/2]Xt and n2 " 3/2, etc. The total mass flow rate is

mb "Xn

i"0

niDi "12

D0 & D1 &32

D2 & + + + " #1:077& + + +$Xt (45)

This flow rate is significantly greater than that of the fractal-likedesign of Fig. 6a. The numbers of canopies of smaller scales thatwould complete the construction of Fig. 6b are n3 " 6, n4 " 6,n5 " 6, n6 " 6, y, but their contributions to the global flow rate (mb)are minor.

The important aspect of the comparison between Fig. 6a and bis that there is a choice [(b) is better than (a)], because each treecontributes to the global flow rate in proportion to its length scale.Had the construction been based simply on the ability to ‘‘fill thearea’’ by repeating an assumed algorithm, as in fractal (spacefilling) practice (e.g., West et al., 1999), there would have been nodifference between (a) and (b), because the triangular area is thesame in both cases, and both designs cover the area. Furthermore,the fractal-like design (a) is simpler and more regular, while thebetter design (b) is strange, and seemingly random.

One may ask, why should (b) look different than (a), and whyshould (b) have three large scales (D0, D1, D2) instead of just one?There is nothing strange about the evolution of the drawing (intime) from (a) to (b). This is the time arrow of the constructal law.It may be possible to find triangular designs that are (marginally)

better than (b), but that should not be necessary in view of theglobal picture that will be discussed in relation to Figs. 8–10.

Discs arranged in a square pattern also cover an areacompletely. One can draw and evaluate the square equivalent ofFig. 6a and by replacing the Xt triangle with a square of side Xs. Theresult is Fig. 7a. The numbers of discs of decreasing scales(D0bD1;D2; . . .$ present on this square will be n0 " 1, n1 " 1,n2 " 4, etc. The performance of this regular (fractal-like) designwill be significantly inferior to that of the square pattern shown inFig. 7b, which is the square equivalent of Fig. 6b. The canopy sizesand numbers in the square design are D0 " 2!1/2Xs and n0 " 2,D1 " (1!2!1/2)Xs and n1 " 2, etc. The total mass flow rateextracted from the Xs-square is

ms "X1

i"0

niDi " 2D0 & 2D1 & 8D2 & + + + " 2:608Xs (46)

Coincidentally, one can show that the m values of Fig. 7a and bform the same ratio (namely 0.71) as the m values of Fig. 6a and b.

Finally, we compare Eq. (46) with Eq. (45) to decide whetherthe square design (Fig. 7b) is better than the triangular design ofFig. 6b. The area is the same in both designs, therefore Xt/Xs "2/31/4 and Eqs. (45) and (46) yield

mb

ms" 0:826 (47)

The square design is better, but not by much. Random effects(geology, climate) will make the distribution of multi-scale treesswitch back and forth between triangle and square and maybehexagon, creating in this way multi-scale patterns that appear

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Fig. 6. Multi-scale canopies projected on the forest floor: (a) triangular pattern with algorithm-based generation of smaller scales and (b) triangular pattern with morelarge-size canopies.

Fig. 7. Square pattern of canopy assemblies: (a) algorithm-based generation of smaller scales and (b) more numerous large-scale canopies for greater ground-air flowconductance.



even more random than the triangle alone, the square alone, andthe hexagon alone. The key feature, however, is that the design iswith multiple scales arranged hierarchically, and that this sort ofdesign is demanded by the constructal law of generating ground-air flow access.

The hierarchical character of the large and small trees of theforest is revealed in Fig. 8, where we plotted the size (Di) and rankof the canopies shown in Fig. 6a and b. The calculation of the rankis explained in Table 1. The largest canopy has the rank 1, and afterthat the canopies are ordered according to size, and countedsequentially. For example, the canopies of size D2 in Fig. 6b aretied for places 4–6. The sizes were estimated graphically byinscribing a circle in the respective curvilinear triangle in whichthe projected canopy would fit.

The data collected for designs (a) and (b) in Table 1 aredisplayed as canopy size versus the canopy rank in Fig. 8. To onevery large canopy belongs an entire ‘‘organization’’, namely twocanopies of next (smaller) size, followed by increasingly largernumbers of progressively smaller scales. This conclusion isreinforced by Fig. 9, which in combination with Table 2summarizes the ranking of scales visible in the square arrange-ments of canopies drawn in Fig. 7a and b. There are no significantdifferences between Figs. 8 and 9.

The noteworthy feature is the alignment of these data asapproximately straight lines on the log–log field of Figs. 8 and 9.

A bird’s eye view of this hierarchy is presented in Fig. 10. This typeof alignment is associated empirically with the Zipf distribution,and it was discovered theoretically in the constructal theory of the

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Fig. 8. The hierarchical distribution of canopy sizes versus rank in the triangularforest floor designs of Fig. 6.

Table 1Sizes, numbers and ranks for the multi-scale canopies populating the forestdesigns of Fig. 6

i Size, Di/Xt 2ni Rank

(a) (b) (a) (b) (a) (b)

0 1 0.789 1 1 1 11 0.155 0.366 2 2 2, 3 2, 32 0.0613 0.211 6 3 4–9 4–63 0.0325 0.054 6 12 10–15 7–244 0.0206 0.024 12 12 16–27 25–365 0.02 0.021 6 12 28–33 37–486 0.0106 0.019 12 12 34–45 49–60

Fig. 9. The hierarchical distribution of canopy sizes versus rank in the squareforest floor designs of Fig. 7.

Table 2Sizes, numbers and ranks for the multi-scale canopies populating the square forestdesign of Fig. 7

i Size, Di/Xs ni Rank

(a) (b) (a) (b) (a) (b)

0 1 0.707 1 2 1 1, 21 0.414 0.3 1 2 2 3, 42 0.107 0.076 4 8 3–6 5–123 0.048 0.036 4 8 7–10 13–204 0.040 0.029 8 16 11–18 21–36

Fig. 10. The Zipfian distribution of canopy sizes versus rank, as a summary ofFigs. 8 and 9.



distribution of multi-scale human settlements on a large territory(Bejan, 2006, pp. 774–779).

6. Discussion

More theoretical progress can be made along this route if weask additional questions about the flow of water through the treeand into the atmosphere. The flow path constructed thus farconsists of channels (root, trunk, branches). This constructioncan be continued toward smaller branches, in the same way as inFig. 5, where we used the trunk and canopy design to deduce thedesign of the branch and canopy design. This step can be repeateda few times, toward smaller scales.

The water stream _m flows through this structure from the baseof the trunk, P(L), to the smallest branches, P0. From the inside ofthe smallest branches to the atmosphere (where the water vaporpressure is Pv), the stream _m must diffuse across a large surfacethat is wrinkled and packed into the interstices formed betweenbranches (this is a model for the main path of water loss, throughthe variable-aperture stomata on leaf surfaces, which provide lowresistance for water loss by diffusion when fully open). This,diffusion at the smallest scales, optimally balanced with hier-archical channels at larger scales, is the tree architecture ofconstructal theory (Bejan, 1997, 2000). It was recognized earlier inhill slope seepage and river channels, alveolar diffusion andbronchial airways, diffusion across capillaries and blood flowthrough arteries and veins, walking and riding on a vehicle inurban traffic, etc. This balance between diffusion and channeling,which fills the volume completely, is why the constructal trees arenot fractal: if one magnifies a subvolume, one sees an image thatis not a repeat of the original image.

Inside the tree canopy, the large surface through whichchanneled _m makes contact with the flowing atmosphere isprovided by leaves that ride on the smallest branches. If theirtotal surface area is A, then the global flow rate crossing A is

_m " c2A#P0 ! Pv$ (48)

where c2 is proportional to the leaf-air mass transfer coefficient,assumed known. In a stronger wind, c2 is larger and can becalculated based on boundary layer mass transfer theory.

The corresponding shorthand expression for _m traveling alongthe trunk and branches is, cf. Eq. (43):

_m " c1L)P#x " L$ ! P0* (49)

Here, we wrote L instead of Dc(x " L), because Dc(x " L) isproportional to L, cf. Eq. (29). Eliminating P0 between Eqs. (48)and (49) we determine the global flow conductance C, from thebase of the trunk to the atmosphere:

C "_m

P#x " L$ ! Pv"

1c1L&

1c2A

! "!1

(50)

Let VT represent the total volume in which the tree resides. Thevolume fractions occupied by wood (trunk and branches), andleaves and air are, respectively, w and l such that w+l " 1. In anorder of magnitude sense, the length scales of the wood and leafvolumes are (wVT)1/3 and (lVT)1/3. Because the leaves are flat, theirarea scales as (lVT)2/3. Together, these scales mean that Eq. (50)becomes

C(1

c1#wVT $1=3&

1

c2#lVT $2=3

" #!1

(51)

where VT is the tree size and V1=3T its length scale (e.g., trunk base

diameter, or height).In conclusion, the global conductance C is proportional to the

tree length scale V1=3T raised to a power between 1 and 2. This is

confirmed by a review of published measurements (Tyree, 2003)of global transpiration in sugar maple (Acer saccharum) of trunkbase diameters in the range 1.3 mm–10 cm, which showed aproportionality between C and #V1=3

T $1:42. Further support for this

conclusion is provided by measurement reported by Ryan et al.(2000) for ponderosa pine (Pinus ponderosa) of two sizes, 12 and36 m high. The measurements show that under various time-dependent conditions (diurnal and seasonal) the length-specificwater flux [i.e., C/(length)2] for 12 m trees is approximately twiceas large as the water flux for 36 m trees. This means that the ratioC(36 m)/C(12 m) is essentially constant in time and equal to 2. Thisalso means that the exponent in the proportionality between Cand #V1=3

T $p is approximately p " 1.37, which is in good agreement

with Tyree (2003) and the discussion of Eq. (51).The balance between diffusion at the smallest (interstitial)

scale and channeling at larger scales, which was demonstrated forseveral classes of tree-shaped flows (e.g., Reis et al., 2004; Miguel,2006), means that there must be an optimal allocation of leafvolume to wood (xylem) volume, so that C is maximum (thexylem volume—the specialized layer of tissue through whichwater flows—is proportionally a fraction of the total woodvolume). Indeed, if we replace l with (1!w) in Eq. (51), we find

C(c1c2VT w1=3#1!w$2=3

c1w1=3V1=3T & c2#1!w$2=3V2=3

T

(52)

The conductance is zero when there are no branches and trunk(w " 0), and when there are no leaves (w " 1). The conductance ismaximum in between. The optimal wood volume fraction isobtained by solving qC/qw " 0, or, in view of the order ofmagnitude character of this analysis, by simply intersecting thetwo asymptotes of C, cf. Eq. (51). This method yields

w

#1!w$2(

c2

c1

! "3

VT (53)

The conclusion is that there is an optimal way to allocate woodvolume to leaf and air volume, and the volume fraction wincreases almost in proportion with (c2/c1)3VT. Larger trees musthave more wood per unit volume than smaller trees. Trees of thesame size (VT) must have a larger wood volume fraction in windyclimates, because c2 increases with the wind speed V.

The relationship between c2 and V is monotonic and can bepredicted based on the analogy between mass transfer andmomentum transfer (Bejan, 2004, pp. 534–536). If V is smallenough, so that the Reynolds number based on leaf lengthscale y is small, Re " Vy/no104, the boundary layers on the leavesare laminar, and the mass transfer coefficient (or c2) is proportionalto Re1=2. This means that c2 is proportional to V1/2. In the oppositeextreme, the entire assembly of leaves is a rough surface withturbulent flow in the fully turbulent and fully rough regime, like theflow of water in a rocky river bed. The skin friction coefficient Cf isconstant (independent of Re), and the corresponding mass transfercoefficient hm is provided by the Colburn analogy for mass transfer,hm=V " #1=2$Cf Pr!2=3; constant. This shows that in the high-V limitthe mass transfer coefficient (or c2) increases as V.

The analysis that brought us to these conclusions is consistentwith analytical definitions and results used in forestry research(e.g., Tyree and Ewers, 1991; Horn, 2000). A well-establishedprinciple is the Huber rule, which relates the leaf specificconductivity (LSC) to the specific conductivity of the stem (ks):

LSC " HV' ks (54)

where HV is the Huber value, defined as the sapwood cross-section (AW) divided by the leaf area distal to the stem (AL):

HV "AW

AL(55)

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In terms of the variables used in this paper, the specificconductivity of the stem and the leaf specific conductivity are

ks "_m

A2W #dP=dz$

(56)

LSC "_m

AW AL#dP=dz$(57)

Combined, Eqs. (55)–(57) reproduce the Huber rule. The presentanalysis goes one step further, because with the optimization thatled to Eq. (53) it provides an additional equation with which toestimate an optimal value for HV.

In summary, it is possible to put the emergence of tree-likearchitectures on a purely theoretical basis, from root to forest. Keyis the integrative approach to understanding the emergence of flowdesign in nature, in line with constructal theory and Turner’s(2007) view that the living flow system is everything, the flow andits environment. In the present case, trees and forests are viewedas integral components (along with river basins, atmosphericcirculation and aerodynamic raindrops) in the global design thatfacilitates the cyclical flow of water in nature. This approach led tothe most basic macroscopic characteristics of tree and forestdesign, and to the discovery, from theory alone, of the principlethat underlies some of the best known empirical correlations oftree water flow performance, e.g., Tyree (2003) and Ryan et al.(2000).

To illustrate the reach of the method that we have used, we endwith another connection between this work and known andaccepted empirical correlations. One example is the well-knownself-thinning law of plant spatial packing, where the meanbiomass of the plant increases as a power law as the number ofplants of the same size decreases (Adler, 1996). A recent review(West and Brown, 2004) showed that the number of trees (Ni) thathave the same linear size (e.g., Di) has been found empirically toobey the proportionality Ni(D!1

i . The same proportionality isfound for multi-scale patches (fragments) of forests, e.g., Fig. 2 inKeitt et al. (1997). This proportionality is sketched with circles inFig. 11. The corresponding rank (Ri) of the trees correlated asNi(D!1

i is calculated by arranging all the trees in the order ofdecreasing sizes, from the largest (k " 1) to the trees of size i:

Ri "Xi

k"1

Nk#Dk$ (58)

The resulting ordering of the empirically correlated trees isindicated with black squares in Fig. 11. The Di!Ri data occupy anarrow strip that has a slope between !1 and !1/2, just like thestrips deduced from the constructal law in Figs. 8–10. This

coincidence suggests that the success of empirical correlationsbetween numbers and sizes of trees is another indication that thetheoretical distribution of tree rankings (e.g., Fig. 10) is correct,and that the single principle on which this entire paper is based isvalid.

We are very grateful for the extremely insightful commentsprovided by the anonymous reviewers, which expanded the rangeof predictions made based on the constructal law in this paper.Their comments deserve serious discussion and future theoreticalwork, however, we use this opportunity to begin the discussionright here:

(i) One comment was that it is not surprising that trees andforests exhibit morphologies that provide access for waterflow, but generalizing this to a holistic architecture involvingtrees and atmospheric circulation seems much less obvious.In reality, our work proceeded the other way around. Severalauthors had the general principle (the constructal law) inmind, and with it they predicted with pencil and paper themorphologies of global water flow as river basins (e.g., Reis,2006), corals and plant roots (Miguel, 2006), atmosphericcirculation and climate (Reis and Bejan, 2006), animal bodymass flow as ‘‘locomotion’’ (Bejan and Marden, 2006), etc.There is great diversity in this list of design predictions,ranging from the biosphere to the atmosphere and thehydrosphere, and covering all the known length and massscales. Early on in the emerging field of constructal theory(e.g., Bejan, 2000) it was considered obvious that the riverdelta too is a flow-access design for point-area flow,predictable based on the same principle, as a river basinturned inside out.Put together, the designs of river basins, deltas and flow ofanimal mass are facilitating the flow of water all over theglobe. The same is happening in the atmosphere and theoceans, because of the patterned circulation known suc-cinctly as ‘‘climate’’. The summarizing question came last:what ‘‘design’’ facilitates the water-flow connection betweenthe land based designs and the atmosphere? Vegetation isone design, for ground-air flow access. Aerodynamic dropletsare another, for air-ground water access (see Fig. 1).This is a new and rich direction of theoretical inquiry inwhich to use the constructal law. There may be othermorphological features of the biosphere that can be predictedand brought in line with the ‘‘holistic architecture’’ of thewater circuit in nature.

(ii) Another comment was to speculate on how the flowarchitecture would change if the facilitating of the watercycle is not true. First, all we have is the well-known circuitthat water executes in nature, and now this paper in whichwe linked in very simple terms the tree-like architecture tothe water-access function coupled with the wind resistancefunction. The generation of vegetation architecture is drivenby more than two objectives (see the first paragraph ofSection 1), but the two drivers are enough for speculating assuggested by the reviewer. If vegetation is not demanded andshaped by the rest of nature (the environmental flows) to putthe ground water back in the air, then, based on our analysis,fixed-mass structures that must withstand the winds will allresemble the Eiffel Tower, not the botanical tree (cf., Fig. 4). Inreality, vegetation is tree-shaped above and below ground,shaped like all the other point-area and point-volume flowsthat facilitate flow access.It is the tree shape that argues most loudly in favor of waterflow access as the raison d’etre of vegetation everywhere. Thismission comes wrapped in the strength of materials question

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Fig. 11. Empirical numbers of canopies of the same size (Ni), and the ranks (Ri) ofsuch canopies.



of how to protect mechanically (and with fixed biomass) thetree-shaped conduits between ground and air. The designsolution is to endow the tree with round, tapered and, aboveeverything, long trunks and branches.Tree size ultimately means rate of rainfall, because the treelength scale is proportional to the rate of water mass flowfacilitated by the tree. The fixed-mass structure must stretchinto the air as high and as wide as possible, and not snap inthe wind. This is how the design arrives at illustrating for usthe universal tendency of trees to bulk up in stressedsubvolumes, and to distribute stresses uniformly throughtheir entire volume. To be able to put the ‘‘axiom of uniformstresses’’ (a solid mechanics design principle) under the sametheoretical roof as the minimization of global flow resistance(a fluid mechanics design principle) is a fundamentaldevelopment in the theory of design in nature.

(iii) Would this be much different if raindrops were spherical andnot aerodynamically shaped? No, in fact drops start outspherical, and all sorts of random effects conspire to preventthem from falling in the way (aerodynamically) in which theywould otherwise tend to fall. Things would be marginallydifferent if all the raindrops would be spherical, however, thesame random effects will prevent this uniformity of shape tooccur. The global flow performance (i.e., the rate of rainfall) isextremely robust to changes and variations in the morphologiesof the individuals. We have seen this in several domainsinvestigated based on constructal theory, from the cross-sectional shapes of river channels to the movement of peoplein urban design. Global features of flow design and flowperformance go hand in hand with the overwhelming diversityexhibited by the individuals that make up the whole.

Determinism and randomness find a home under the sametheoretical tent. In fact, the tree architecture is an illustration (anicon) of this duality. Pattern is discernible from a distance, so thatit appears simple enough to be grasped by the mind. Diversity(chance) is discernible close up. There is no contradiction betweenthe two, just harmony in how the individuals contribute to andbenefit from the global flow.

Along this holistic line, we rediscover the tree as an individualshaped by the forest, and the forest as an individual shaped by therest of the global flowing environment (Fig. 1).

Acknowledgments

This research was supported by the Air Force Office of ScientificResearch based on a grant for ‘‘Constructal Technology forThermal Management of Aircraft’’. Jaedal Lee’s work at DukeUniversity was sponsored by the Korea Research Foundation GrantMOEHRD, KRF-2006-612-D00011.

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