Unifying constructal theory of tree roots, canopies and forests

ARTICLE IN PRESS Journal of Theoretical Biology 254 (2008) 529–540

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Unifying constructal theory of tree roots, canopies and forests A. Bejan a,, S. Lorente b, J. Lee a a b

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA 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 in fo

abstract

Article history: Received 30 October 2007 Received in revised form 14 June 2008 Accepted 27 June 2008 Available online 2 July 2008

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 Leonardo’s rule, Huber’s rule, Zipf’s distribution, and the Fibonacci sequence. The difference between modeling (description) and theory (prediction) is brought into evidence. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Constructal theory Design in nature Roots Trees Forests Leonardo’s rule Fibonacci sequence Zipf distribution Eiffel Tower

1. Introduction

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

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 survive strong 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  Corresponding author. Tel.: +1 919 660 5314; fax: +1 919 660 8963.

E-mail address: [email protected] (A. Bejan). 0022-5193/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2008.06.026

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).

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Nomenclature a, b a0 A AB AL At AW c1, c2 C CD D Dc Dc,B DL Dt Dt,B F0 h HV It kr ks Kr Kx, Kz L

factors, Eqs. (8), (9), (23), (28) and (29) factor, Eq. (25) area (m2) branch cross-section at the trunk (m2) leaf area distal to stem (m2) tree cross-section at x (m2) sapwood cross-section (m2) factors, Eqs. (48) and (49) global flow conductance, Eq. (50) drag coefficient diameter (m) canopy diameter (m) diameter of branch canopy (m) diameter at z ¼ L (m) trunk diameter (m) diameter of branch (m) drag force per unit length (N/m) frustum height (m) Huber value area moment of inertia (m4) radial specific conductivity m/(s Pa) stem specific conductivity m/(s Pa) radial permeability (m2/s) longitudinal permeability (m2/s) length (m)

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 solidification (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 evolutionary principle of configuration generation for greater flow access 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 flowing through the plant, from the ground to the atmosphere.

LB LSC m, n m _ m _B m p P Pg PL Pv P0 Ri sm u uB v V V VT w x, z Xs Xt

m n r

branch length (m) leaf specific conductivity exponents, Eqs. (8), (9) and (23) bending moment (N m) mass flow rate (kg/s) branch mass flow rate (kg/s) exponent pressure (Pa) ground pressure (Pa) pressure at z ¼ L (Pa) vapor pressure (Pa) branch tip pressure (Pa) rank of trees of size Di maximum bending stress (N/m2) Darcy (volume averaged) longitudinal velocity (m/s) branch Darcy longitudinal velocity (m/s) Darcy radial velocity (m/s) wind speed (m/s) volume (m3) total volume (m3) wood volume fraction longitudinal coordinates (m) side of square (m), Fig. 7b side of equilateral triangle (m), Fig. 6b viscosity (kg/s m) kinematic viscosity (m2/s) density (kg/m3)

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 morphology, 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 it provides 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: Z L p 2 V¼ D dz (1) 0 4 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 fibers that are oriented longitudinally.

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and r is the density of water. Eqs. (4) and (5) yield d 2 ðD uÞ ¼ 4vD dz

(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: _L¼r m

p 4

D2 ðLÞuðLÞ

(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 ¼ 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

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.

We assume that the (L, D) body is sufficiently slender, so that the pressure inside the body depends mainly on longitudinal position, Pðr; zÞ ffi PðzÞ. 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 K z dP u¼ m dz

(2)

where m is the fluid viscosity. Because of the Pðr; zÞ ffi PðzÞ assumption, for the transversal volume averaged velocity v (oriented toward negative r) we write approximately: vffi

K r P g  PðzÞ m 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 permeabilities Kz and Kr are two constants. The radial permeability Kr should not be confused with the concept of radial water conductivity kr, which is defined as the ratio between the radial flux 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 ¼ 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 _ ¼ rpDv dz dm

(4)

_ is the longitudinal mass flow rate at level z: where m _ ¼r m

p 4

D2 u

(5)

Pg  PðzÞ ¼ azn m=K z

(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): m¼1

(10)

2

b nðn þ 1Þ ¼ 8

Kr Kz

(11)

The volume constraint (1) yields a third condition: 2

b L3 ¼

12

p

V

(12)

A fourth condition follows from the statement that the overall pressure difference is fixed, which in view of Eq. (9) means that Pg  PL ¼ aLn ; m=K z

constant

(13)

Finally, the mass flow rate through the z ¼ L end of the body is, cf. Eq. (7):   p K dðP g  PÞ p 2 _ L ¼ r ðbLÞ2 z m ¼ r b anLnþ1 (14) dx 4 m 4 z¼L for which b(n) and L(n) are furnished by Eqs. (11) and (12). The resulting ground-level flow rate is 1=3    p K 2=3 12 n1=3 _ L ¼ r ðaLn Þ 8 r m V (15) 4 Kz p ðn þ 1Þ2=3 with the observation that (aLn) is a constant, cf. Eq. (13). _ L depends on root shape (n) according to the In conclusion, m function n1/3/(n+1)2/3. This function is maximum when n¼1

(16)

Working back, we find that the constructal root design must have this length and aspect ratio:   3VK z 1=3 (17) L¼ pK r   L 1 K z 1=2 ¼ DL 2 K r

(18)

The constructal root shape is conical. The slenderness of this cone is dictated by the anisotropy of the porous structure (Kz/Kr)1/2.

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The root is more slender when the vascular structure is more permeable longitudinally. Another important feature of the discovered root geometry is that the longitudinal volume averaged fluid velocity (u) is independent of longitudinal position (z), because n ¼ 1 means that dP/dz ¼ constant, and u¼

K z Pg  PL m L

_L¼r m

p 4

ð0ozoLÞ

ðbLÞ2 u ¼

  K r 2=3 ðPg  PL Þð3VÞ1=3 p n Kz

Kz

(19)

(20)

The morphological implications of this theoretical feature are important. If the porous structure is a bundle of tiny capillary channels, then the fluid velocity through each tube must be constant, and must not depend on the size of the root crosssection (p/4)D2(z) that the channel pierces. On the other hand, earlier work on constructal design (Bejan and Lorente, 2004) has shown the following: flow strangulation is not good for flow performance, the constructal configuration of a long capillary with specified flow rate and volume is the one where the crosssection does not vary with longitudinal position, and the crosssection is round. Combining this with the new conclusion that u must not depend on z, we discover the internal structure of the constructal root body. The longitudinal tubes must be round, with diameters that do not vary with z, even though some tubes are longer than other tubes. The external shape and internal structure of the root body discovered in this section are sketched on the right side of Fig. 2. Another feature of the constructal root design is visible in Eq. (3). Because both (PgP) and D are proportional to z, we conclude that v must also be z-independent. One can show that  1=2 v Kr ¼ (21) u Kz The anisotropy of the vascular porous structure dictates the ratio between constant-v and constant-u, in the same way that it dictates the root slenderness ratio DL/L, cf. Eq. (18). There is a considerable body of literature on the modeling of water flow through roots, and a common assumption is that the root is a porous conduit with constant diameter. The analysis then yields a pressure that varies nonlinearly (exponentially) along the conduit (Landsbert and Fowkes, 1978; Frensch and Stendle, 1989; Roose and Fowler, 2004). This is consistent with the analysis shown in this section, for if this analysis is repeated by postulating that D is constant, then in place of the present conclusion (dP/ dz ¼ constant) we find that P(z) must vary exponentially. However, measurements made on root segments show that the pressure does not vary linearly (Zwieniecki et al., 2003; McElrone et al., 2004). This is a good opportunity for next-generation analytical and experimental work. For example, the Landsbert and Fowkes (1978) modeling should be combined with the optimal tapering of the conduits, i.e., with the constructal law of facilitating flow access by allowing the flow geometry to morph.

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

by the wind on the canopy as a problem of two-dimensional flow, in a horizontal plane that cuts the trunk and the canopy. The trunk and the canopy are modeled as two bodies of revolution, with unknown diameters Dt(x) and Dc(x), where x is measured downward from the top of the tree. The drag force per unit length (x) experienced by the tree canopy is F 0 ¼ C D Dc  12rV 2

where V is the horizontal wind speed and Dc(x) is the radius of the canopy at the distance x from the tree top. We assume that the Reynolds number VDc/n is greater than 103, so that the drag coefficient CD is a constant approximately equal to 1. To give our search for geometry sufficient generality, assume that the canopy has a shape that belongs in the family of powerlaw functions: Dc ¼ axn

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

(23)

where a is a constant and the shape exponent n is not known. The bending moment experienced by the trunk at the distance x from the tree top is Z x a0 xnþ2 MðxÞ ¼ (24) F 0 ðx  xÞ dx ¼ ðn þ 1Þðn þ 2Þ 0 where a0 is another constant: a0 ¼

a rV 2 C D 2

(25)

We now turn our attention to the maximum bending stresses in the cross-section of the trunk of diameter Dt(x): sm ¼

MðxÞ Dt ðxÞ It ðxÞ 2

(26)

where It ¼ pD4t =64. The stress sm occurs in the dorsal and ventral fibers of the trunk, as the trunk bends in the wind that pushes the canopy. ‘‘Optimal distribution of imperfection’’ (Bejan, 2000) means that sm must be the same over the entire height of the tree. According to Eq. (24), the trunk diameter must vary as Dt ðxÞ ¼

3. Trunk and canopy shape

(22)



32a0 =p sm ðn þ 1Þðn þ 2Þ

1=3

xðnþ2Þ=3

(27)

This is an important result, but it is not the end of the story. It says that if we know the canopy shape (n), then we can predict the trunk shape, and vice versa (Fig. 4). To determine the trunk and canopy shapes uniquely, we need an additional idea (Section 4). If the canopy is shaped as a cone (n ¼ 1), then the trunk is also shaped as a cone, Dt/x ¼ constant. Fig. 4 shows that if the canopy has a round top (e.g., n ¼ 1/2) then the trunk diameter must vary as Dt/x5/6 ¼ constant, which is not much different than Dt/ x ¼ constant. If the canopy has a very sharp tip (e.g., n ¼ 3/2), then Dt(x) must vary as Dt/x7/6 ¼ constant, which again is not far

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

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 MðxÞ=D3t . The conclusion is that the trunk (or solitary pole) is the strongest to bending when it is conical, n ¼ 1. The same result follows from the subsequent discussion of Eq. (27), if we assume Dc ¼ Dt. A famous structure that only now reveals its bendingresistance 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 ‘‘imperfection’’ (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 of West 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 shaved off. 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 the soil, 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 ¼ 1. The conical shape is also the constructal choice for the large and progressively smaller lateral branches, provided that their mechanical design is dominated by wind resistance considerations, not by the

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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  1=2 Dt ðxÞ Kr ¼2 ¼b (28) x Kx

tip of the trunk (x ¼ 0). The pressure at the tip of the branch LB is also P0. In accordance with Eq. (19), we write u¼

K x PðxÞ  P 0 m x

uB ¼

K x;B PðxÞ  P 0 m LB

(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) and Dc(x) is provided by Eq. (27) with n ¼ 1, in combination with Eqs. (25), (28) and (29): Dc ðxÞ a 3psm K r ¼ ¼ Dt ðxÞ b 2C D rV 2 K x

(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 V50 km/h (14 m/s). Eq. (30) with CD1 yields Dc/Dt2.42  106(Kr/Kx)walnut and, after additional algebra, 3=2 Dc =Ltrunk 4:8  106 ðK r =K x Þwalknut . The corresponding estimates for a pine tree (Pinus silvestris) with sm ’ 6:6  107 N/m2 in a windy climate with V100 km/h (28 m/s) are Dc/Dt3.4  105(Kr/ 3=2 Kx)pine and Dc =Ltrunk 6:8  105 ðK r =K x Þpine . 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 ¼ b) and canopy (Dc/ x ¼ a) also hold for the branch LB(x) located at level x: Dt;B ¼ b; LB

Dc;B ¼a LB

(31)

Furthermore, in accordance with Eq. (29) for the canopy, Dc(x) is the same as 2LB(x), which means that LB ðxÞ ¼ 12ax

(32)

Dt;B ðxÞ ¼ 12abx

(33)

Dc;B ðxÞ ¼ 12a2 x

(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 trunk segment is _ B ¼ rvpDt h m

u K x LB ¼ uB K x;B x

p 4

D2t;B

h ¼ 14ax ¼ 14LB

(41)

In conclusion, the vertical segment of trunk (h) that is responsible for the flow rate into one lateral branch is proportional 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 ¼ aLB, cf. Eq. (34). Comparing h with Dc,B, we find that hðxÞ 1 ¼ Dc;B ðxÞ 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 massconservation analysis that led to Eq. (41). The discovery is that Leonardo’s rule (e.g., Horn, 2000; Shinozaki et al., 1964) is deducible from Eq. (41), in these steps. The trunk cross-sectional 2 area at the distance x from the tip is At ðxÞ ¼ ðp=4Þb x2 . At the top 2 2 of the h frustum, it is At ðx  hÞ ¼ ðp=4Þb ðx  hÞ . The reduction in trunk cross-sectional area from x to xh is DAt ¼ At ðxÞ  At ðx  hÞ. The cross-sectional area of the thick end of the single branch 2 allocated to h is AB ¼ ðp=4ÞD2t;B ¼ ðp=4Þb L2B . The ratio between the decrease in trunk cross-sectional area and the branch crosssectional area allocated to that decrease is, after some algebra, DAt =AB ¼ ð2=aÞ½1  ða=8Þ. In view of Eq. (42), where ð1=2aÞ1, according to constructal theory the ratio DAt =AB must be a constant of order 1.

(35)

(36)

where Dt,B is the diameter of branch LB at the junction with the _ B between Eqs. (35) and (36), and using Eqs. trunk. Eliminating m (28), (32) and (33), we find that h is proportional to x: (37)

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

(40)

It is reasonable to assume that the longitudinal permeability of the wood to be the same in the trunk and the branch, K x ffi K x;B , such that Eq. (37) reduces to

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

h uB a2 ¼ x u 8

(39)

which yield

Dc ðxÞ ¼a x

_ B ¼ ruB m

(38)

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

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The area ratio would have been exactly 1 according to Leonardo’s 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 real tree, just like Leonardo’s 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 they fill the ‘‘alveoli’’ created in the canopy ‘‘cone’’ by two counterrotating 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 Leonardo’s 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. This prediction 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 compression (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 xindependent sm) has the form D ¼ ax2, where a is constant. Such a branch has zero thickness in the vicinity of the tip (dD/dx ¼ 0 at x ¼ 0+), and is not a shape found in nature. This result alone

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indicates that the tips of branches are not shaped by weight loading alone, and that wind loading (which prescribes D ¼ 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 ¼ 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: _ ¼ ru m

p 4

D2t ðx ¼ LÞ ¼

p b2 K x ½Pðx ¼ LÞ  P0 Dc ðx ¼ LÞ 4 an

(43)

where x ¼ L indicates ground level and Dc(x ¼ 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 _ and Dc(x ¼ L). This also means that the total mass flow between m rate is proportional to the tree height L. This proportionality will be modified somewhat when we take into account the additional _ as it flows from the smallest flow resistance encountered by m 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) _ i ), where each m _ i is is an area covered with fluid mass sources (m 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 triangle is Xt, then the diameter of the largest canopy disc is D0 ¼ Xt, and the number of D0-size canopies present on one Xt triangle is n0 ¼ 1/2. For the next smaller canopy, the parameters are D1 ¼ (31/21/2)Xt and n1 ¼ 1. At the next smaller size, the number of canopies is n2 ¼ 3, and the disc size is D2 ¼ 0.0613Xt. The construction continues in an infinite number of steps (n3 ¼ 3, n4 ¼ 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 ¼

1 X i¼0

ni Di ¼

1 D0 þ D1 þ 3D2 þ    ¼ ð0:761 þ   ÞX t 2

(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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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.

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 ¼ [(31/2+1)/2]Xt and n0 ¼ 1/2, D1 ¼ [(31/21)/2]Xt and n1 ¼ 1, D2 ¼ [(131/2)/2]Xt and n2 ¼ 3/2, etc. The total mass flow rate is mb ¼

n X i¼0

ni Di ¼

1 3 D0 þ D1 þ D2 þ    ¼ ð1:077 þ   ÞX t 2 2

(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 ¼ 6, n4 ¼ 6, n5 ¼ 6, n6 ¼ 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. 8–10. 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 Xt triangle with a square of side Xs. The result is Fig. 7a. The numbers of discs of decreasing scales (D0 bD1 ; D2 ; . . .Þ present on this square will be n0 ¼ 1, n1 ¼ 1, n2 ¼ 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 ¼ 21/2Xs and n0 ¼ 2, D1 ¼ (121/2)Xs and n1 ¼ 2, etc. The total mass flow rate extracted from the Xs-square is ms ¼

1 X

ni Di ¼ 2D0 þ 2D1 þ 8D2 þ    ¼ 2:608X s

(46)

i¼0

Coincidentally, one can show that the m values of Fig. 7a and b form the same ratio (namely 0.71) as the m values of Fig. 6a and b. Finally, we compare Eq. (46) with Eq. (45) to decide 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 ¼ 2/31/4 and Eqs. (45) and (46) yield mb ¼ 0:826 ms

(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 maybe hexagon, creating in this way multi-scale patterns that appear

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

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

0 1 2 3 4 5 6

Size, Di/Xt

2ni

Fig. 9. The hierarchical distribution of canopy sizes versus rank in the square forest 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

Rank

(a)

(b)

(a)

(b)

(a)

(b)

1 0.155 0.0613 0.0325 0.0206 0.02 0.0106

0.789 0.366 0.211 0.054 0.024 0.021 0.019

1 2 6 6 12 6 12

1 2 3 12 12 12 12

1 2, 3 4–9 10–15 16–27 28–33 34–45

1 2, 3 4–6 7–24 25–36 37–48 49–60

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 groundair 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 counted sequentially. For example, the canopies of size D2 in Fig. 6b are tied for places 4–6. 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 arrangements 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 as approximately straight lines on the log–log field of Figs. 8 and 9.

537

0 1 2 3 4

Size, Di/Xs

ni

Rank

(a)

(b)

(a)

(b)

(a)

(b)

1 0.414 0.107 0.048 0.040

0.707 0.3 0.076 0.036 0.029

1 1 4 4 8

2 2 8 8 16

1 2 3–6 7–10 11–18

1, 2 3, 4 5–12 13–20 21–36

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

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

6. Discussion More theoretical progress can be made along this route if we ask additional questions about the flow of water through the tree and 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. _ flows through this structure from the base The water stream m of the trunk, P(L), to the smallest branches, P0. From the inside of the smallest branches to the atmosphere (where the water vapor _ must diffuse across a large surface pressure is Pv), the stream m 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 hierarchical 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 _ makes contact with the flowing atmosphere is channeled m provided by leaves that ride on the smallest branches. If their total surface area is A, then the global flow rate crossing A is _ ¼ c2 AðP 0  P v Þ m

(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. _ traveling along The corresponding shorthand expression for m the trunk and branches is, cf. Eq. (43): _ ¼ c1 L½Pðx ¼ LÞ  P 0  m

(49)

Here, we wrote L instead of Dc(x ¼ L), because Dc(x ¼ 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:   _ m 1 1 1 þ C¼ ¼ (50) Pðx ¼ LÞ  P v c1 L c2 A 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 ¼ 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 " #1 1 1 C þ (51) c1 ðwV T Þ1=3 c2 ðlV T Þ2=3 1=3

where VT is the tree size and V T its length scale (e.g., trunk base diameter, or height). In conclusion, the global conductance C is proportional to the 1=3 tree length scale V T 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 mm–10 cm, which showed a 1=3 proportionality between C and ðV T Þ1:42 . 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 timedependent conditions (diurnal and seasonal) the length-specific water 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 ratio C(36 m)/C(12 m) is essentially constant in time and equal to 2. This also means that the exponent in the proportionality between C 1=3 and ðV T Þp is approximately p ¼ 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 volume—the specialized layer of tissue through which water flows—is proportionally a fraction of the total wood volume). Indeed, if we replace l with (1w) in Eq. (51), we find C

c1 c2 V T w1=3 ð1  wÞ2=3 1=3 c1 w1=3 V T

2=3

þ c2 ð1  wÞ2=3 V T

(52)

The conductance is zero when there are no branches and trunk (w ¼ 0), and when there are no leaves (w ¼ 1). The conductance is maximum in between. The optimal wood volume fraction is obtained by solving qC/qw ¼ 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  3 w c2  VT (53) 2 c1 ð1  wÞ The conclusion is that there is an optimal way to allocate wood volume to leaf and air volume, and the volume fraction w increases 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. 534–536). If V is small enough, so that the Reynolds number based on leaf length scale y is small, Re ¼ Vy/no104, the boundary layers on the leaves are 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 is constant (independent of Re), and the corresponding mass transfer coefficient hm is provided by the Colburn analogy for mass transfer, hm =V ¼ ð1=2ÞC f Pr 2=3 ; constant. This shows that in the high-V limit 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 ¼ HV  ks

(54)

where HV is the Huber value, defined as the sapwood crosssection (AW) divided by the leaf area distal to the stem (AL): HV ¼

AW AL

(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 ¼

_ m A2W ðdP=dzÞ

LSC ¼

_ m AW AL ðdP=dzÞ

(56)

(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 Turner’s (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 forest design, 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 D1 i . The same proportionality is found for multi-scale patches (fragments) of forests, e.g., Fig. 2 in Keitt et al. (1997). This proportionality is sketched with circles in Fig. 11. The corresponding rank (Ri) of the trees correlated as Ni D1 is calculated by arranging all the trees in the order of i decreasing sizes, from the largest (k ¼ 1) to the trees of size i: Ri ¼

i X

N k ðDk Þ

(58)

k¼1

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. 8–10. This

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

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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), atmospheric circulation 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 succinctly 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 to the 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 d’eˆtre of vegetation everywhere. This mission comes wrapped in the strength of materials question

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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 in the 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, the same 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 crosssectional 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 Lee’s work at Duke University was sponsored by the Korea Research Foundation Grant MOEHRD, KRF-2006-612-D00011. References Adler, F.R., 1996. A model of self-thinning through local competition. Proc. Natl. Acad. Sci. USA 93, 9980–9984. Bejan, A., 1997. Advanced Engineering Thermodynamics, second ed. Wiley, New York, pp. 798–804. Bejan, A., 2000. Shape and Structure from Engineering to Nature. Cambridge University Press, Cambridge, UK.

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