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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 2007

Received in revised form

14 June 2008

Accepted 27 June 2008

Keywords:

Constructal theory

Design in nature

Roots

Trees

Forests

Leonardos rule

Fibonacci sequence

Zipf distribution

Eiffel 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 unifying

theoretical basis, which is provided by the constructal law. Key is the integrative approach to

understanding the emergence of designedness in nature. Trees and forests are viewed as integral

components (along with dendritic river basins, aerodynamic raindrops, and atmospheric and oceanic

circulation) 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 wood

mass raised to 1/3, the proportionality between total water mass flow rate and tree length, the

proportionality between the tree flow conductance and the tree length scale raised to a power between

1 and 2, the existence of forest floor plans that maximize ground-air flow access, the proportionality

between the length scale of the tree and its rank raised to a power between 1 and 1/2, and the

inverse proportionality between the tree size and number of trees of the same size. This paper further

shows that there exists an optimal ratio of leaf volume divided by total tree volume, trees of the same

size must have a larger wood volume fraction in windy climates, and larger trees must pack more wood

per unit of tree volume than smaller trees. Comparisons with empirical correlations and formulas based

on ad hoc models are provided. This theory predicts classical notions such as Leonardos rule, Hubers

rule, Zipfs 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 complex

evolutionary process. The generation of the tree architecture is

driven by many competing demands. The tree must catch

sunlight, absorb CO2 and put water into the atmosphere, while

competing for all these flows with its neighbors. The tree must

survive droughts and resist pests. It must adapt, morph and grow

toward 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 are

higher. It must be able to distribute its stresses as uniformly as

possible, so that all its fibers work hard toward the continued

survival of the mechanical structure.

On the background of this complexity in demands and

functionality, two demands stand out. The tree must facilitate

the flow of water, and must be strong mechanically. The demand

to pass water is made abundantly clear by the strong geographical

correlation between the density (and sizes) of trees and the rate of

rainfall (Fig. 1). It is also made clear by the dendritic architecture,

ARTICLE IN PRESS

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 the

circuit executed by water on the globe. Examples of such flow configurations are

aerodynamic droplets, tree-shaped river basins and deltas, vegetation, and all

forms 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.: +1 919 660 5314; fax: +1919660 8963.

E-mail address: [email protected] (A. Bejan).

Journal of Theoretical Biology ] (]]]]) ]]]]]]

Please cite this article as: Bejan, A., et al., Unifying constructal theory of tree roots, canopies and forests. J. Theor. Biol. (2008),doi:10.1016/j.jtbi.2008.06.026
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which is the best way to provide flow access between one point

and a finite-size volume (Bejan, 1997). The demand to be strong

mechanically is made clear by features such as the tapered trunks

and limbs with round cross-section, and other design-like features

identified in this article. These features of designedness in solid

structures facilitate the flow of stresses, which is synonymous with

mechanical strength.

According to constructal theory, plants (vegetation) occur and

survive in order to facilitate ground-air mass transfer (Bejan,

2006, p. 770). Recently, constructal theory (Bejan, 1997, 2000) has

shown that dendritic crystals such as snowflakes are the most

effective heat-flow configurations for achieving rapid solidifica-

tion (Bejan, 1997; Ciobanas et al., 2006). The same mental viewing

was used to explain the variations in the morphology of stony

corals and bacterial colonies and the design of plant roots (Miguel,

2006; Biondini, 2008). The 23-level architecture of the lung (Reis

et 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 predict

the generation of natural configuration across the board, from

biology to geophysics and social dynamics (for reviews, see Bejan

and Lorente, 2006; Bejan, 2006; Bejan and Merkx, 2007).

In this paper, we rely on constructal theory in order to construct

based on a single principle the main features of plants, from root and

canopy to forest. We take an integrative approach to trees as live flow

systems that evolve as components of the larger whole (the

environment). We regard the plant as a physical flow architecture

that evolves to meet two objectives: maximum mechanical strength

against 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 of

our work is to demonstrate that the existence of tree-like

architecture can be anticipated as a mental viewing based on the

constructal law. The work is purely theoretical. Although

comparisons with natural forms are made, the work is not

intended to describe and correlate empirically the diversity of

plant measurements found in nature. Although we are not nearly

as familiar as our biology colleagues with the sequence of

theoretical and empirical advances made on vegetation morphol-

ogy, in constructal theory we have a physics method with which

we have predicted natural flow design across the board (Bejan and

Lorente, 2006). We bring to this table of discussion the tools of

strength of materials, fluid mechanics, and, above all, the

engineering thinking of multi-objective design. We believe that

our physics work will be of interest because of its engineering

origins 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 above

ground, into the trunk of the plant. The ground water enters the

root through all the points of its surface. In the simplest possible

description, the root is a porous solid structure shaped as a body

of revolution (Fig. 2). The shape of the body [L, D(z)] is not known,

but the volume is fixed:

V

ZL0

p

4D2 dz (1)

The flow of water through the root body is in the Darcy regime.

The permeability of the porous structure in the longitudinal

direction (Kz) is greater than the permeability in the transversal

direction (Kr). Anisotropy is due to the fact that the woody

vascular tissue (the xylem) is characterized by vessels and fibersthat are oriented longitudinally.

ARTICLE IN PRESS

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 (m

2)

AL leaf area distal to stem (m2

)At tree cross-section at x (m

2)AW sapwood cross-section (m

2)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 (m

2/s)Kx, Kz longitudinal permeability (m

2/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 Dism maximum bending stress (N/m

2)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 (m

3)w

wood volume fractionx, z longitudinal coordinates (m)

Xs side of square (m), Fig. 7bXt side of equilateral triangle (m), Fig. 6b

m viscosity (kg/s m)n kinematic viscosity (m2/s)r density (kg/m3)

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We assume that the (L, D) body is sufficiently slender, so that

the pressure inside the body depends mainly on longitudinal

position, Pr;z ffi Pz. This slenderness assumption is analogous

to the slender boundary layer assumption in boundary layer

theory. For Darcy flow, the z volume averaged longitudinal

velocity is given by

u Kzm

dP

dz(2)

where m is the fluid viscosity. Because of the Pr;z ffi Pzassumption, for the transversal volume averaged velocity v

(oriented toward negative r) we write approximately:

v ffiKrm

Pg Pz

D=2(3)

The definition of the radial permeability (Kr) of the root body

as a Darcy porous medium is Eq. (3). This definition is consistent

with Eq. (2), which is the definition of the longitudinal

permeability of the root as a nonisotropic Darcy porous

medium (e.g., Nield and Bejan, 2006). The directional permeabil-

ities Kz and Kr are two constants. The radial permeability Krshould not be confused with the concept of radial water

conductivity kr, which is defined as the ratio between the radialflux of water and the radial pressure difference [e.g., Eq. (3.3) in

Roose and Fowler, 2004].

The ground-water pressure (Pg) outside the body is assumed

constant. This means that in this model the hydrostatic pressure

variation with depth Pg(z) is assumed to be negligible, and that the

root sketched in Fig. 2 can have any orientation relative to gravity.

Ground level is indicated by z L: here the pressure is PL, and is

lower than Pg. Throughout the body, P(z) is lower than Pg, and the

radial 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

d

dzD2u 4vD (6)

Summing up, the three Eqs. (2), (3) and (6) should be sufficient

for 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 must

determine the shape [L, D(z)] that allows the global pressure

difference (PgPL) to pump the largest flow rate of water to the

ground level:

_mL rp

4D2LuL (7)

subject to the volume constraint (1). Instead of trying a numerical

approach or one based on variational calculus, here we use a much

simpler 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 assumption

that the function P(z) belongs to the family represented by

Pg Pz

m=Kz az

n

(9)

where a and n are two additional constants. When we substitute

assumptions (8) and (9) into Eqs. (2) and (3), and then substitute

the resulting u and v expressions into Eq. (6), we obtain

two compatibility conditions for the assumptions made in

Eqs. (8) and (9):

m 1 (10)

b2nn 1 8KrKz

(11)

The volume constraint (1) yields a third condition:

b2L3 12

pV (12)

A fourth condition follows from the statement that the overall

pressure difference is fixed, which in view of Eq. (9) means that

Pg PLm=Kz

aLn; constant (13)

Finally, the mass flow rate through the z L end of the body is, cf.

Eq. (7):

_mL rp

4bL2

Kzm

dPg P

dx

zL

rp

4b2anLn1 (14)

for which b(n) and L(n) are furnished by Eqs. (11) and (12). The

resulting ground-level flow rate is

_mL rp

4

aLn 8Kr

Kz

2=3 12

p

V 1=3 n1=3

n 12=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 have

this length and aspect ratio:

L 3VKzpKr

1=3(17)

L

DL

1

2

KzKr

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.

ARTICLE IN PRESS

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 ] (]]]]) ]]]]]] 3



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off the conical trunk shape. In sum, we have discovered that the

shape of the trunk that is uniformly stressed is relatively

insensitive to how the canopy is shaped. A conical trunk is

essentially a uniform-stress body in bending for a wide variety of

canopy shapes that deviate (concave vs. convex) from the conical

canopy shape sketched in Fig. 3.

A simpler version of the problem solved in this section is to

search for the optimal shape of the trunk Dt(x) when there

is no canopy. The trunk alone is the obstacle in the wind,

and its bending is due to the distributed drag force F0 of

Eq. (22), in which Dc is replaced by Dt. The analysis leads to

Eq. (27) where M(x) varies as xn+2, and sm (constant) is

proportional to Mx=D3t. The conclusion is that the trunk (or

solitary pole) is the strongest to bending when it is conical, n 1.

The same result follows from the subsequent discussion of

Eq. (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). The

shape of the structure is not conical (Fig. 4) because in addition to

bending in the wind, the structure must be strong in compression.

The optimization of tower shape for uniform distribution of

compressive stress leads to a tower profile that becomes

exponentially narrower with altitude. The shape of a tower that

is uniformly resistant to lateral bending and axial compression is

between the conical and the exponential. This apparent im-

perfection (deviation from the exponential) of the Eiffel Tower

has been a puzzle until now (see the end of Section 4).This discussion of the Eiffel Tower also sheds light on a major

mechanical difference between the present theory and the model

ofWest et al. (1999). In the present work, the mechanical function

is to resist bending due to horizontal wind drag, as in the upper

section of the Eiffel Tower. In the model of West et al., the

mechanical function is to resist buckling under its own weight, on

the vertical. Of course, all modes of resisting fracture are

important, but, which is the more important? Buckling is not,

because the weight of the tree is static, totally independent of the

notoriously random and damaging behavior of the flowing

environment. The wind is much more dangerous. Record breaking

wind speeds make news all over the globe, and their combined

effect can only be one: the cutting of the trunks, branches and

leaves 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-ending

assault.

4. Conical trunks, branches and canopies

The preceding section unveiled the architecture of a tree that

has evolved, so that its stresses flow best and its maximum

allowable stress is distributed uniformly. This tree supports the

largest load (i.e., it resists the strongest wind) when the tree

volume is specified. Conversely, the same architecture withstands

a specified load (wind) by using minimum tree volume. In

summary, the multitude of near-conical designs discovered in

Eq. (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 from

thick to thin, along the trunk and its branches.

There is no question that the maximization of access for fluid

flow plays a major role in the configuring of the tree. This is why

the tree is tree-shaped, dendritic, one trunk with branches, and

branches with many more smaller branches. How do the designs

of Eq. (27) facilitate the maximization of access for fluid flow?

The answer is provided by the constructal root discovered in

Section 2 and Eqs. (17)(21). The constructal shape for a body

permeated by Darcy flow with two permeabilities (Kz, Kr) is

conical. The longitudinal and lateral seepage velocities (u, v) are

uniform, independent of the longitudinal position z. For a root, the

lateral seepage is provided by direct (contact) diffusion from thesoil, and indirect seepage from root branches, rootlets and root

hairs. For the tree trunk above the ground, the lateral flow that

accounts for v is facilitated (ducted) almost entirely by lateral

branches. Above the ground, the lateral v is concentrated discretely

in branches that are distributed appropriately along and around

the trunk (see the discussion of the Fibonacci sequence at the end

of this section).

The theoretical step that we make here is this: the constructal

flow design of the root is the same as the flow design of the trunk

and canopy. From this we deduce that out of the multitude of

near-conical trunk shapes for wind resistance, Eq. (27), the

constructal law selects the conical shape, n 1. The conical shape

is also the constructal choice for the large and progressively

smaller lateral branches, provided that their mechanical design isdominated by wind resistance considerations, not by the

ARTICLE IN PRESS

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




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resistance to their own body weight. We return to this observation

in the last paragraph of this section.

Recognition of the conical trunk and canopy shapes means that

the analysis in this section begins with Eqs. (18) and (23), which

for the tree trunk and canopy reduce to

Dtx

x 2

KrK

x

1=2

b (28)

Dcx

x a (29)

Here, it should be noted that for the tree trunk the axial

coordinate (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):

Dcx

Dtx

a

b

3psm

2CDrV2

KrKx

(30)

Eq. (30) recommends a large Dc/Dt ratio for trees with hard wood

in moderate winds, and a small Dc/Dt ratio for trees with soft

wood in windy climates. A hard-wood example is the walnut tree

( Juglans regia) with sm 1:2 108 N/m2, in a mild climate

represented by V$50 km/h (14 m/s). Eq. (30) with CD$1 yields

Dc/Dt$2.42106(Kr/Kx)walnut and, after additional algebra,

Dc=Ltrunk$4:8 106Kr=Kx

3=2walknut

. The corresponding estimates

for a pine tree (Pinus silvestris) with sm 6:6 107 N/m2 in a

windy climate with V$100 km/h (28 m/s) are Dc/Dt$3.4105(Kr/

Kx)pine and Dc=Ltrunk$6:8 105Kr=Kx

3=2pine.

How many branches should be placed in the canopy, and at

what 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) is

the same as 2LB(x), which means that

LBx 12ax (32)

Dt;Bx 12abx (33)

Dc;Bx 12a

2x (34)

A single branch LB(x) resides in a frustum of the conical

canopy: the frustum height is h(x) and the base radius is LB(x).

In the center of this frustum, there is a trunk segment (another

conical frustum) of height h(x) and diameter Dt(x). The trunk

frustum 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)

IfuB is the longitudinal fluid velocity along the branch LB, then the

same fluid mass flow rate can be written as

_mB ruBp

4D2t;B (36)

where Dt,B is the diameter of branch LB at the junction with the

trunk. Eliminating _mB between Eqs. (35) and (36), and using Eqs.

(28), (32) and (33), we find that h is proportional to x:

h

x

uBu

a2

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 is

also P0. In accordance with Eq. (19), we write

u Kxm

Px P0x

(38)

uB Kx;Bm

Px P0LB

(39)

which yield

u

uB

KxKx;B

LBx

(40)

It is reasonable to assume that the longitudinal permeability of

the wood to be the same in the trunk and the branch, Kx ffi Kx;B,

such that Eq. (37) reduces to

h 14ax 14

LB (41)

In conclusion, the vertical segment of trunk (h) that is

responsible for the flow rate into one lateral branch is propor-

tional to the length of the branch. Another dimension that is

proportional to LB(x) is the diameter of the conical branch

canopy circumscribed to the horizontal LB, namely Dc,B aLB, cf.Eq. (34). Comparing h with Dc,B, we find that

hx

Dc;Bx

1

2a(42)

which is a constant of order 1. In other words, there is room in the

global canopy (L, Dc) to install one LB-long branch on every h-tall

segment of tree trunk. The geometrical features discovered in this

section have been sketched in Fig. 5.

One of the reviewers of the original manuscript asked us to

compare this tree architecture with that of the model of West et

al. (1999). This was a great suggestion because it leads to an

important theoretical discovery that is hidden in the mass-

conservation analysis that led to Eq. (41). The discovery is that

Leonardos rule (e.g., Horn, 2000; Shinozaki et al., 1964) isdeducible from Eq. (41), in these steps. The trunk cross-sectional

area at the distance x from the tip is Atx p=4b2x2. At the top

of the h frustum, it is Atx h p=4b2x h2. The reduction in

trunk cross-sectional area from x to xh is DAt Atx Atx h.

The cross-sectional area of the thick end of the single branch

allocated to h is AB p=4D2t;B p=4b

2L2B. 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=a1 a=8. In view of Eq. (42), where 1=2a$1,

according to constructal theory the ratio DAt=AB must be a

constant of order 1.

ARTICLE IN PRESS

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




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The area ratio would have been exactly 1 according to

Leonardos rule, which was based on visual study and drawings

of trees. This rule is predicted here based on the constructal law

and other first principles such as the conservation of water mass

flow rate. In West et al.s (1999) model, this rule was assumed, not

predicted. It was assumed along with several other assumptions

(e.g., the tree-shaped structure), so that the model could become

compact and useful as a facsimile

as a description

of the realtree, just like Leonardos observations. It is because of such

assumptions that the allometric relations derived algebraically

from West et al.s (1999) model are description, not prediction.

This remark is necessary because it contradicts West et al.s use of

the words predicted values in the reporting of their derivations

(e.g., Table 1, p. 667). Additional comments on West et al.s model

are provided by Kozlowski and Konarzewski (2004) and Makela

and Valentine (2006).

In the present paper, the tree architecture and the tapering of

its limbs are deduced from a single postulate which is the

constructal law. Furthermore, because there is one lateral branch

per trunk segment h(x), and because h decreases in proportion

with x, the best way to fill the tree canopy with the canopies of the

lateral 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 arrange

themselves up the trunk, one discovers the Fibonacci sequence

(e.g., Livio, 2002).

Like Leonardos rule, the Fibonacci sequence is the result of

Eq. (42), the predicted conical canopy shape, and the geometric

requirement that the next branch and canopy should shoot laterally

into the space that is impeded the least by the branch canopies

situated immediately above and below. The need of minimum

interference between branches is a restatement of the constructal

law, i.e., the tendency to morph to have greater flow access for water

from ground to wind. Each branch reaches for the pocket of volume

that contains the least humid air flow. This principle is universal, and

is fundamentally different than ad hoc statements such as stems

grow in positions that would optimize their exposure to sun, rain,

and air (Livio, 2002), and phyllotaxis simply represents a state of

minimal 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 point

consists of cones inside cones. The large conical trunk and canopy

hosts a close packing of smaller conical branches and conical branch

canopies. One can take this construction further to smaller scales,

and see the architecture of each branch as a conical canopy packed

with smaller conical branches and their smaller canopies. In such a

construction, the wood volume is a fraction of its total volume, i.e., a

fraction of the volume of the large canopy, which scales as L3. From

this follows the prediction that the trunk length L must be

proportional to the total wood mass raised to the power 1/3. Thisprediction agrees very well with measurements of five orders of

magnitude of tree mass scales (e.g., Table 2 in Bertram, 1989).

In closing, we return to the Eiffel Tower discussed at the end of

the preceding section, where we noted that strength in compres-

sion (under the weight) near the base was combined with

strength in bending (subject to lateral wind) in the upper body

of the tower. This discussion is relevant in the modeling of the

horizontal branch, which in this section was based on the

assumption that the loading is due to lateral wind. The branch

is also loaded in the vertical direction, under its own weight. If we

assume 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. Such

a branch has zero thickness in the vicinity of the tip (d D/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 weight

loading alone, and that wind loading (which prescribes D ax

and finite D at small x) is the more appropriate model there. For

the thick end of the branch, it can be argued that D ax2 is a

realistic shape, and that near the trunk the weight loading of the

beam is the dominant shaping mechanism, just like in the Eiffel

Tower near the ground.

5. Forest

Forests are highly complex systems, and their study has

generated a significant body of literature (for reviews, see Keitt

et al., 1997; Urban et al., 1987). Multi-scale models of landscape

pattern and process are being applied, for example, models with

spatially embedded patch-scale processes (Weishampel and

Urban, 1996). To review this activity is beyond our ability, and is

not our objective. Here we continue on the constructal path traced

up to this point (Fig. 1): if the root, trunk and canopy architecture

is driven by the tendency to generate flow access for water, from

ground to air, then, according to the same mental viewing (i.e.,

according to the same theory), the forest too should have an

architecture that promotes flow access.

The fluid flow rate ducted by the entire tree from the ground to

the tips of the trunk and branches is:

_m rup

4D2tx L

p

4

b2

anKxPx L P0Dcx L (43)

where x L indicates ground level and Dc(x L) is the diameter of

the canopy projected as a disc on the ground. The important

feature of the tree design discovered so far is the proportionality

between _m and Dc(x L). This also means that the total mass flow

rate is proportional to the tree height L. This proportionality will

be modified somewhat when we take into account the additional

flow resistance encountered by _m as it flows from the smallest

branches (P0) through the leaves and into the atmosphere (Pa). See

Section 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 is

proportional to the diameter of the circular area allocated to it.

From the constructal law of generating ground-to-air fluid flow

access follows the design of the forest.

The principle is to morph the area into a configuration with mass

sources (or disc-shaped canopy projections) such that the total fluid

flow rate lifted from the area is the largest. From this invocation of

the constructal law follows, first, the prediction that the forest must

have trees of many sizes, few large trees interspaced with more and

more numerous smaller trees. This is illustrated in Fig. 6a with a

triangular area covered by canopy projections arranged according to

the algorithm that a single disc is inserted in the curvilinear triangle

that 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 the

number of D0-size canopies present on one Xt triangle is n0 1/2.

For the next smaller canopy, the parameters are D1 (31/21/2)Xt

and n1 1. At the next smaller size, the number of canopies isn2 3, and the disc size is D2 0.0613Xt. The construction

continues in an infinite number of steps (n3 3, n4 6, y) until

the Xt triangle is covered completely. The image that would result

from this infinite compounding of detail would be a fractal. The total

fluid flow rate vehicled by the design from the triangular area of Fig.

6a is proportional to

ma X1i0

niDi 1

2D0 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,

ARTICLE IN PRESS




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a better forest architecture is the one where the larger discs are more

numerous. This observation leads to Fig. 6b, where the Xt triangle

is covered more uniformly by larger discs, in this sequence: D0

[(31/2+1)/2]Xt and n0 1/2, D1 [(31/21)/2]Xt and n1 1, D2

[(131/2)/2]Xt and n2 3/2, etc. The total mass flow rate is

mb Xni0

niDi 1

2D0 D1

3

2D2 1:077 Xt (45)

This flow rate is significantly greater than that of the fractal-like

design of Fig. 6a. The numbers of canopies of smaller scales that

would 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 b

is that there is a choice [(b) is better than (a)], because each tree

contributes to the global flow rate in proportion to its length scale.

Had the construction been based simply on the ability to fill the

area by repeating an assumed algorithm, as in fractal (space

filling) practice (e.g., West et al., 1999), there would have been no

difference between (a) and (b), because the triangular area is the

same in both cases, and both designs cover the area. Furthermore,

the fractal-like design (a) is simpler and more regular, while the

better design (b) is strange, and seemingly random.

One may ask, why should (b) look different than (a), and why

should (b) have three large scales (D0, D1, D2) instead of just one?

There is nothing strange about the evolution of the drawing (in

time) 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 the

global picture that will be discussed in relation to Figs. 810.

Discs arranged in a square pattern also cover an area

completely. One can draw and evaluate the square equivalent of

Fig. 6a and by replacing the Xttriangle with a square of side Xs. The

result 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) design

will be significantly inferior to that of the square pattern shown in

Fig. 7b, which is the square equivalent of Fig. 6b. The canopy sizes

and numbers in the square design are D0 21/2Xs and n0 2,

D1 (121/2)Xs and n1 2, etc. The total mass flow rate

extracted from the Xs-square is

ms X1i0

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

Coincidentally, one can show that the m values of Fig. 7a and b

form the same ratio (namely 0.71) as the m values ofFig. 6a and b.

Finally, we compare Eq. (46) with Eq. (45) to decide whether

the square design (Fig. 7b) is better than the triangular design of

Fig. 6b. The area is the same in both designs, therefore Xt/Xs

2/31/4 and Eqs. (45) and (46) yield

mbms

0:826 (47)

The square design is better, but not by much. Random effects

(geology, climate) will make the distribution of multi-scale trees

switch back and forth between triangle and square and maybehexagon, creating in this way multi-scale patterns that appear

ARTICLE IN PRESS

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 more

large-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 flow

conductance.




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even more random than the triangle alone, the square alone, and

the hexagon alone. The key feature, however, is that the design is

with multiple scales arranged hierarchically, and that this sort of

design is demanded by the constructal law of generating ground-

air flow access.

The hierarchical character of the large and small trees of the

forest is revealed in Fig. 8, where we plotted the size (Di) and rank

of the canopies shown in Fig. 6a and b. The calculation of the rank

is explained in Table 1. The largest canopy has the rank 1, and after

that the canopies are ordered according to size, and countedsequentially. For example, the canopies of size D2 in Fig. 6b are

tied for places 46. The sizes were estimated graphically by

inscribing a circle in the respective curvilinear triangle in which

the projected canopy would fit.

The data collected for designs (a) and (b) in Table 1 are

displayed as canopy size versus the canopy rank in Fig. 8. To one

very large canopy belongs an entire organization, namely two

canopies of next (smaller) size, followed by increasingly larger

numbers of progressively smaller scales. This conclusion is

reinforced by Fig. 9, which in combination with Table 2

summarizes the ranking of scales visible in the square arrange-

ments of canopies drawn in Fig. 7a and b. There are no significant

differences between Figs. 8 and 9.

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

A birds eye view of this hierarchy is presented in Fig. 10. This type

of alignment is associated empirically with the Zipf distribution,and it was discovered theoretically in the constructal theory of the

ARTICLE IN PRESS

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

Table 1

Sizes, numbers and ranks for the multi-scale canopies populating the forest

designs of Fig. 6

i Size, Di/Xt 2ni Rank

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

0 1 0.789 1 1 1 1

1 0.155 0.366 2 2 2, 3 2, 3

2 0.0613 0.211 6 3 49 46

3 0.0325 0.054 6 12 1015 724

4 0.0206 0.024 12 12 1627 2536

5 0.02 0.021 6 12 2833 37486 0.0106 0.019 12 12 3445 4960

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

Table 2

Sizes, numbers and ranks for the multi-scale canopies populating the square forest

design of Fig. 7

i Size, Di/Xs ni Rank

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

0 1 0.707 1 2 1 1, 2

1 0.414 0.3 1 2 2 3, 4

2 0.107 0.076 4 8 36 512

3 0.048 0.036 4 8 710 1320

4 0.040 0.029 8 16 1118 2136

Fig. 10. The Zipfian distribution of canopy sizes versus rank, as a summary of

Figs. 8 and 9.




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distribution of multi-scale human settlements on a large territory

(Bejan, 2006, pp. 774779).

6. Discussion

More theoretical progress can be made along this route if we

ask additional questions about the flow of water through the treeand into the atmosphere. The flow path constructed thus far

consists of channels (root, trunk, branches). This construction

can be continued toward smaller branches, in the same way as in

Fig. 5, where we used the trunk and canopy design to deduce the

design of the branch and canopy design. This step can be repeated

a few times, toward smaller scales.

The water stream _m flows through this structure from the base

of the trunk, P(L), to the smallest branches, P0. From the inside of

the smallest branches to the atmosphere (where the water vapor

pressure is Pv), the stream _m must diffuse across a large surface

that is wrinkled and packed into the interstices formed between

branches (this is a model for the main path of water loss, through

the variable-aperture stomata on leaf surfaces, which provide low

resistance 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 of

constructal theory (Bejan, 1997, 2000). It was recognized earlier in

hill slope seepage and river channels, alveolar diffusion and

bronchial airways, diffusion across capillaries and blood flow

through arteries and veins, walking and riding on a vehicle in

urban traffic, etc. This balance between diffusion and channeling,

which fills the volume completely, is why the constructal trees are

not fractal: if one magnifies a subvolume, one sees an image that

is not a repeat of the original image.

Inside the tree canopy, the large surface through which

channeled _m makes contact with the flowing atmosphere is

provided by leaves that ride on the smallest branches. If their

total surface area is A, then the global flow rate crossing A is

_m c2AP0 Pv (48)

where c2 is proportional to the leaf-air mass transfer coefficient,

assumed known. In a stronger wind, c2 is larger and can be

calculated based on boundary layer mass transfer theory.

The corresponding shorthand expression for _m traveling along

the trunk and branches is, cf. Eq. (43):

_m c1LPx L P0 (49)

Here, we wrote L instead of Dc(x L), because Dc(x L) is

proportional to L, cf. Eq. (29). Eliminating P0 between Eqs. (48)

and (49) we determine the global flow conductance C, from the

base of the trunk to the atmosphere:

C

_m

Px L Pv

1

c1L

1

c2A 1

(50)

Let VT represent the total volume in which the tree resides. The

volume fractions occupied by wood (trunk and branches), and

leaves and air are, respectively, w and l such that w+l 1. In an

order of magnitude sense, the length scales of the wood and leaf

volumes are (wVT)1/3 and (lVT)

1/3. Because the leaves are flat, their

area scales as (lVT)2/3. Together, these scales mean that Eq. (50)

becomes

C$1

c1wVT1=3

1

c2lVT2=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 thetree length scale V

1=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 trunk

base diameters in the range 1.3 mm10 cm, which showed a

proportionality 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 and

36 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 twice

as large as the water flux for 36 m trees. This means that the ratioC(36m)/C(12m) is essentially constant in time and equal to 2. This

also means that the exponent in the proportionality between C

and V1=3T

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 for

several classes of tree-shaped flows (e.g., Reis et al., 2004; Miguel,

2006), means that there must be an optimal allocation of leaf

volume to wood (xylem) volume, so that C is maximum (the

xylem volumethe specialized layer of tissue through which

water flowsis proportionally a fraction of the total wood

volume). Indeed, if we replace l with (1w) in Eq. (51), we find

C$c1c2VTw

1=31 w2=3

c1w1=3V1=3T c21 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 is

maximum in between. The optimal wood volume fraction is

obtained by solving qC/qw 0, or, in view of the order of

magnitude character of this analysis, by simply intersecting the

two asymptotes of C, cf. Eq. (51). This method yields

w

1 w2$

c2c1

3VT (53)

The conclusion is that there is an optimal way to allocate wood

volume to leaf and air volume, and the volume fraction wincreases almost in proportion with (c2/c1)

3VT. Larger trees must

have more wood per unit volume than smaller trees. Trees of the

same size (VT) must have a larger wood volume fraction in windy

climates, because c2 increases with the wind speed V.

The relationship between c2 and V is monotonic and can be

predicted based on the analogy between mass transfer and

momentum transfer (Bejan, 2004, pp. 534536). If V is small

enough, so that the Reynolds number based on leaf length

scale y is small, Re Vy/no104, the boundary layers on the leavesare laminar, and the mass transfer coefficient (or c2) is proportional

to Re1=2. This means that c2 is proportional to V1/2. In the opposite

extreme, the entire assembly of leaves is a rough surface with

turbulent flow in the fully turbulent and fully rough regime, like the

flow of water in a rocky river bed. The skin friction coefficient Cf isconstant (independent of Re), and the corresponding mass transfer

coefficient hm is provided by the Colburn analogy for mass transfer,hm=V 1=2CfPr

2=3; constant. This shows that in the high-Vlimit

the mass transfer coefficient (or c2) increases as V.

The analysis that brought us to these conclusions is consistent

with analytical definitions and results used in forestry research

(e.g., Tyree and Ewers, 1991; Horn, 2000). A well-established

principle is the Huber rule, which relates the leaf specific

conductivity (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 AWAL

(55)

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In terms of the variables used in this paper, the specific

conductivity of the stem and the leaf specific conductivity are

ks _m

A2WdP=dz(56)

LSC _m

AWALdP=dz

(57)

Combined, Eqs. (55)(57) reproduce the Huber rule. The present

analysis goes one step further, because with the optimization that

led to Eq. (53) it provides an additional equation with which to

estimate an optimal value for HV.

In summary, it is possible to put the emergence of tree-like

architectures on a purely theoretical basis, from root to forest. Key

is the integrative approach to understanding the emergence of flow

design in nature, in line with constructal theory and Turners

(2007) view that the living flow system is everything, the flow and

its environment. In the present case, trees and forests are viewed

as integral components (along with river basins, atmospheric

circulation and aerodynamic raindrops) in the global design that

facilitates the cyclical flow of water in nature. This approach led to

the most basic macroscopic characteristics of tree and forestdesign, and to the discovery, from theory alone, of the principle

that underlies some of the best known empirical correlations of

tree water flow performance, e.g., Tyree (2003) and Ryan et al.

(2000).

To illustrate the reach of the method that we have used, we end

with another connection between this work and known and

accepted empirical correlations. One example is the well-known

self-thinning law of plant spatial packing, where the mean

biomass of the plant increases as a power law as the number of

plants of the same size decreases (Adler, 1996). A recent review

(West and Brown, 2004) showed that the number of trees (Ni) that

have the same linear size (e.g., Di) has been found empirically to

obey the proportionality Ni$D1i . The same proportionality is

found for multi-scale patches (fragments) of forests, e.g., Fig. 2 inKeitt et al. (1997). This proportionality is sketched with circles in

Fig. 11. The corresponding rank (Ri) of the trees correlated asNi$D

1i is calculated by arranging all the trees in the order of

decreasing sizes, from the largest (k 1) to the trees of size i:

Ri Xik1

NkDk (58)

The resulting ordering of the empirically correlated trees is

indicated with black squares in Fig. 11. The DiRi data occupy a

narrow strip that has a slope between 1 and 1/2, just like the

strips deduced from the constructal law in Figs. 810. This

coincidence suggests that the success of empirical correlations

between numbers and sizes of trees is another indication that the

theoretical distribution of tree rankings (e.g., Fig. 10) is correct,

and that the single principle on which this entire paper is based is

valid.

We are very grateful for the extremely insightful comments

provided by the anonymous reviewers, which expanded the range

of predictions made based on the constructal law in this paper.Their comments deserve serious discussion and future theoretical

work, however, we use this opportunity to begin the discussion

right here:

(i) One comment was that it is not surprising that trees and

forests exhibit morphologies that provide access for water

flow, but generalizing this to a holistic architecture involving

trees and atmospheric circulation seems much less obvious.

In reality, our work proceeded the other way around. Several

authors had the general principle (the constructal law) in

mind, and with it they predicted with pencil and paper the

morphologies 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 body

mass 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 the

hydrosphere, and covering all the known length and mass

scales. Early on in the emerging field of constructal theory

(e.g., Bejan, 2000) it was considered obvious that the river

delta too is a flow-access design for point-area flow,

predictable based on the same principle, as a river basin

turned inside out.

Put together, the designs of river basins, deltas and flow of

animal mass are facilitating the flow of water all over the

globe. The same is happening in the atmosphere and the

oceans, because of the patterned circulation known suc-

cinctly as climate. The summarizing question came last:

what design facilitates the water-flow connection between

the land based designs and the atmosphere? Vegetation is

one design, for ground-air flow access. Aerodynamic droplets

are another, for air-ground water access (see Fig. 1).

This is a new and rich direction of theoretical inquiry in

which to use the constructal law. There may be other

morphological features of the biosphere that can be predicted

and brought in line with the holistic architecture of the

water circuit in nature.

(ii) Another comment was to speculate on how the flow

architecture would change if the facilitating of the water

cycle is not true. First, all we have is the well-known circuit

that water executes in nature, and now this paper in which

we linked in very simple terms the tree-like architecture tothe water-access function coupled with the wind resistance

function. The generation of vegetation architecture is driven

by more than two objectives (see the first paragraph of

Section 1), but the two drivers are enough for speculating as

suggested by the reviewer. If vegetation is not demanded and

shaped by the rest of nature (the environmental flows) to put

the ground water back in the air, then, based on our analysis,

fixed-mass structures that must withstand the winds will all

resemble the Eiffel Tower, not the botanical tree (cf., Fig. 4). In

reality, vegetation is tree-shaped above and below ground,

shaped like all the other point-area and point-volume flows

that facilitate flow access.

It is the tree shape that argues most loudly in favor of water

flow access as the raison detre 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) of

such canopies.




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of how to protect mechanically (and with fixed biomass) the

tree-shaped conduits between ground and air. The design

solution is to endow the tree with round, tapered and, above

everything, long trunks and branches.

Tree size ultimately means rate of rainfall, because the tree

length scale is proportional to the rate of water mass flow

facilitated by the tree. The fixed-mass structure must stretch

into the air as high and as wide as possible, and not snap inthe wind. This is how the design arrives at illustrating for us

the universal tendency of trees to bulk up in stressed

subvolumes, and to distribute stresses uniformly through

their entire volume. To be able to put the axiom of uniform

stresses (a solid mechanics design principle) under the same

theoretical roof as the minimization of global flow resistance

(a fluid mechanics design principle) is a fundamental

development in the theory of design in nature.

(iii) Would this be much different if raindrops were spherical and

not aerodynamically shaped? No, in fact drops start out

spherical, and all sorts of random effects conspire to prevent

them from falling in the way (aerodynamically) in which they

would otherwise tend to fall. Things would be marginally

different if all the raindrops would be spherical, however, thesame random effects will prevent this uniformity of shape to

occur. The global flow performance (i.e., the rate of rainfall) is

extremely robust to changes and variations in the morphologies

of the individuals. We have seen this in several domains

investigated based on constructal theory, from the cross-

sectional shapes of river channels to the movement of people

in urban design. Global features of flow design and flow

performance go hand in hand with the overwhelming diversity

exhibited by the individuals that make up the whole.

Determinism and randomness find a home under the same

theoretical tent. In fact, the tree architecture is an illustration (an

icon) of this duality. Pattern is discernible from a distance, so that

it appears simple enough to be grasped by the mind. Diversity

(chance) is discernible close up. There is no contradiction between

the two, just harmony in how the individuals contribute to and

benefit from the global flow.

Along this holistic line, we rediscover the tree as an individual

shaped by the forest, and the forest as an individual shaped by the

rest of the global flowing environment (Fig. 1).

Acknowledgments

This research was supported by the Air Force Office of Scientific

Research based on a grant for Constructal Technology for

Thermal Management of Aircraft. Jaedal Lees work at Duke

University was sponsored by the Korea Research Foundation Grant

MOEHRD, KRF-2006-612-D00011.

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