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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: abejan@duke.edu (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
http://www.sciencedirect.com/science/journal/yjtbihttp://www.elsevier.com/locate/yjtbihttp://dx.doi.org/10.1016/j.jtbi.2008.06.026mailto:abejan@duke.eduhttp://dx.doi.org/10.1016/j.jtbi.2008.06.026http://dx.doi.org/10.1016/j.jtbi.2008.06.026mailto:abejan@duke.eduhttp://dx.doi.org/10.1016/j.jtbi.2008.06.026http://www.elsevier.com/locate/yjtbihttp://www.sciencedirect.com/science/journal/yjtbi -
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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)
A. Bejan et al. / Journal of Theoretical Biology ] (]]]]) ]]]]]]2
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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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
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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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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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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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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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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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,
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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
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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 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
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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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