Introduction
 Top of page
 Summary
 Introduction
 Materials and Methods
 Results
 Discussion
 Acknowledgements
 References
 Appendix: Relations between scaling exponents in
Interest in the link between carbon uptake and water loss has yielded numerous efforts to quantify the efficiency of the water transport network of plants (Banavar et al., 1999; McCulloh et al., 2003, 2004; McCulloh & Sperry, 2005a,b; Anfodillo et al., 2006; Weitz et al., 2006; Atala & Lusk, 2008). Much of this work has focused on the xylem tissue of woody stems, where the xylem conduits (vessels or tracheids) may be providing structural support to the plant in addition to transporting water. The additional support task compromises the potential hydraulic efficiency of water transport (McCulloh et al., 2004). Within leaves, though, substantial structural support is provided by hydrostatic pressure and specialized nonvascular tissues such as collenchyma and sclerenchyma. To the extent that this nonxylary structural support holds up the leaf, this would theoretically leave the xylem conduits free to achieve their maximum transport efficiency. Assessing transport efficiency within the lamina is complicated by the typically reticulate networks that are losing water to transpiration (Canny, 1993; McCulloh & Sperry, 2005b). By contrast, it is relatively easy to quantify network efficiency in compound leaves because the vascular tissue can be compared at discrete branching levels, the petiole and petiolule. Using these distinct levels in 13 species, we compared compliance with two theoretical optima that maximize the hydraulic conductance of this simple tworanked vascular network.
The first optimum, Murray's law (ML), predicts how the conduit diameters should change across branching points of a vascular network so as to minimize the power (work/time) driving a given flow rate through a network of fixed total volume (Murray, 1926). As Sherman (1981) demonstrated, a Murray law network also maximizes the hydraulic conductance (volume flow rate per pressure drop). Maximizing conductance (rather than minimizing power at one flow rate) is undoubtedly the real adaptive significance of Murray's law, because a maximumconductance network is optimal at all flow rates (Sherman, 1981). Murray's law was derived for the specific case of the cardiovascular system which delivers blood through a single branching tube. The law states that conductance is maximized when the sum of the conduit radii cubed (Σr^{3}) is conserved at all points along the flow path from aorta to the beginning of the capillaries. Conservation of Σr^{3} requires an increase in the crosssectional area (Σr^{2}) from aorta to capillaries.
Unlike the cardiovascular system for which it was developed, Murray's law does not define the sole hydraulic optimum for the plant vascular system. In the cardiovascular system the branching of the vascular network equals the branching of a single tube. The ratio of the number of daughtertomother tubes (N_{R}) cannot be varied independently of the network branching structure. Murray's law gives the ‘best’ daughtertomother tube taper ratio (D_{R}) for the N_{R} dictated by branching architecture. The evolution of N_{R} is constrained by the need to deliver fluids spatially, and Murray's law D_{R} evolved to maximize the conductance of this mammalian vascular topography.
Xylem networks are more complex because they are composed of many conduits in parallel and in series at every branch level, including the trunk. That means that N_{R} is potentially independent of the branching network. When two daughter arteries join at the aorta, N_{R} = 2. When two petiolules each containing hundreds of conduits join at the petiole, N_{R} does not have to be 2. N_{R} can be any number greater than zero or less than the theoretical maximum for a petiole with just one conduit. Optimization of the network conductance is not just a matter of D_{R}, because N_{R} can also vary independently of the network branching topography.
The fact that there are at least two variables that are potentially free to vary across the branch junctions of plants means that there are at least two theoretical optima for maximizing conductance. The distinction between the two optima is shown graphically in Fig. 1. The contour lines on this figure are hydraulic conductances of a simple network with one petiole and two petiolules. The tubes within each branch rank are of constant diameter, and the total number of petiolule tubes is constant. All other parameters, including the volume of the network, are constant except for N_{R} and D_{R}.
The contours in Fig. 1 describe a diagonal ridge that ascends to greater conductance as N_{R} increases and D_{R} decreases. The ascent of the ridge corresponds to fewer, larger tubes in the petiole. The ridge cannot be ascended indefinitely because of limits on permissible values of D_{R} and N_{R}. The N_{R} reaches its maximum theoretical value when the petiole has just a single tube (N_{R} max on ‘y’ axis; McCulloh et al., 2003). At N_{R}=N_{R }max, the greatest conductance is achieved at the horizontal tangent of the conductance contour where D_{R} conforms to Murray's law (asterisk on ML diagonal in Fig. 1). This global optimum is represented by the cardiovascular system which is a single branched tube that approximates Murray law taper (Sherman, 1981).
Plants are never plumbed with a single branched tube and so are never at their N_{R} max. Negative sap pressures make a singletube network vulnerable to complete failure in the event of a single air leak or cavitation event (Ewers et al., 2007). Greater N_{R} also results in a greater departure of the xylem network from area preservation (Fig. 1; AP diagonal) towards areaincreasing branching at Murray's law. Extreme increases in the crosssectional area of conduits would result in topheavy networks that could be mechanically unstable. Hydraulic and mechanical safety considerations combine to keep the N_{R} of plant xylem well below its theoretical maximum and potentially free to vary independently of the plant's branching system.
The alternative scenario to Murray's law for plant xylem is that D_{R} is more constrained than N_{R} for developmental or functional reasons. Such a constraint could result from a developmental link between vessel diameter and branch rank, or a limit to maximum vessel diameter. In this case, the ‘best’ network would have the minimum permissible D_{R}, and the greatest conductance would be achieved at the vertical tangent of the conductance contour where N_{R} conforms to the ‘YO’ diagonal. Because this optimum is the opposite of Murray's law, we call it the ‘Yarrum’ optimum (YO; Sperry et al., 2008).
Here, we derive both Murray's law and the Yarrum optimum for a simple network composed of the conduits within a petiolule and two daughter petiolules of equal length. The derivations make explicit the distinction between the two optima. Following Sherman's example (1981), the optima are derived from a maximum conductance criterion because this is more biologically relevant than the equivalent minimum power derivation. The series conductance of the network is represented as the reciprocal resistances (R), because they are additive in series. The Hagen–Poiseuille equation predicts:
 (Eqn 1)
where l is the axis length, n is the number of conduits in parallel, and r is the conduit radius, which is assumed constant within a rank. The subscript ‘0’ denotes petiole values, and subscript ‘1’ denotes the totals for the daughter petiolules. The D_{R} is r_{1}/r_{0}. The mean radiuses of petiole or petiolule vessels are assumed to substitute for r_{0} or r_{1}, and mean petiolule vessel diameters are not expected to differ within a compound leaf when leaflets are of approximately similar size. Constant η is the fluid viscosity and k is the factor by which the resistivity exceeds the Hagen–Poiseuille prediction because of endwalls and other obstructions (k ≥ 1). The volume, V, of this network is
 (Eqn 2)
Network volume is proportional to the conduit wall volume if wall thickness is proportional to conduit diameter (McCulloh & Sperry, 2006); the latter is approximately true for xylem of a given cavitation resistance (Hacke et al., 2001). In both derivations, we hold the vascular investment, V, constant and solve for the minimum R under either Murray's law or Yarrum conditions. To derive Murray's law, we hold the numbers of conduits constant (n_{0} and n_{1}) and vary the conduit taper (D_{R}) to find the value that minimizes R. Because volume (V) is constant, we must also allow r_{0} to vary with D_{R}. We use the Lagrange multiplier method to find where R (D_{R}, r_{0}) (Eqn 1) is minimized subject to the constraint that V(D_{R}, r_{0}) = 0, where we have subtracted a constant from Eqn 2 to set V = 0. At the minimum R, the Lagrange multiplier, λ, defines the two equalities (Edwards & Penney, 1998, p. 864):
 (Eqn 3a)
 (Eqn 3b)
From Eqns 1 and 2 the partial derivatives in Eqn 3 are:
 (Eqn 4a)
 (Eqn 4b)
 (Eqn 4c)
 (Eqn 4d)
 (Eqn 5)
 D_{R}= (n_{0}/n_{1})^{1/3}
 D_{R} = N_{R}^{−1/3} (Eqn 6)
for N_{R}=n_{1}/n_{0}. Equation 6 is Murray's law for the special case where tubes within a rank are of equal diameter. However, we have previously shown that conservation of Σr^{3} across branch points is within 3% whether computed from actual vessel distributions or their mean values (McCulloh & Sperry, 2006). Note that the lengths cancel out, indicating that the optimal taper is independent of branch lengths.
To find the Yarrum optimum, we hold the taper constant (D_{R}) and vary the relative numbers of conduits across ranks (varying n_{0} for a constant n_{1}) to find the value that minimizes R. Because volume (V) is constant, we must allow r_{0} to vary with n_{0}. As before, we use the Lagrange multiplier method to find where R(n_{0}, r_{0}) (Eqn 1) is minimized subject to the constraint that V(n_{0}, r_{0}) = 0. At the minimum R, the Lagrange multiplier, λ, defines the two equalities:
 (Eqn 7a)
 (Eqn 7b)
 (Eqn 8a)
 (Eqn 8b)
The derivatives for 7b are 4c and 4d. Substituting 8a and 8b into 7a yields:
 (Eqn 9)
Substituting λ, 4c and 4d into 7b, and simplifying, gives:
 (Eqn 10a)
 (Eqn 10b)
where the branch length ratio L_{R} = l_{1}/l_{0}. Using the quadratic formula to solve for N_{R} yields the following relevant root (the other root gives negative N_{R}; signs in the valid root have been simplified):
 (Eqn 11)
which is the Yarrum optimum. Note that the branch lengths do not cancel out, indicating that the optimal NR depends on the relative lengths of the petiole and petiolule. The shorter the petiolules relative to the petiole (smaller the L_{R}), the fewer conduits they need in parallel (relative to the petiole) to minimize the total resistance (the smaller the optimal N_{R}). As L_{R} approaches zero, the lowest optimal N_{R} converges on 1.41 . For typical D_{R} < 1, Yarrum's N_{R} will always exceed 1.4, meaning that petiolule conduits will always be more numerous than petiole conduits at the Yarrum optimum.
The existence of two hydraulic optima raises the question of what variable is most constrained during evolution and development. If the evolution of N_{R} is more constrained (as in the singletube mammalian cardiovascular system where it is at N_{R} max), then the vascular conductance through the branching system can only be optimized by evolving the value of D_{R} predicted by Murray law. Alternatively, if the D_{R} and L_{R} are more constrained, then the vascular conductance through the branch system would be maximized by evolving the Yarrum optimum N_{R}. To date, the Yarrum optimum has not been recognized, or evaluated, in any system. To address the question of whether leaf vasculature follows Murray's law, or the Yarrum optimum, or conduit area preservation, or none of these patterns, we measured D_{R}, L_{R}, and N_{R} across the petiole vs petiolule ranks of compound leaves and compared them with the predictions. Leaves were studied in six tropical and five temperate angiosperm species, and in two temperate fern species. In addition to evaluating these two optima, we examined the evidence for size and speciesindependent scaling patterns in the hydraulic architecture of leaves.
Results
 Top of page
 Summary
 Introduction
 Materials and Methods
 Results
 Discussion
 Acknowledgements
 References
 Appendix: Relations between scaling exponents in
On the landscape of possible D_{R} vs N_{R} combinations, the species clustered more closely around the ML (Murray's law; Fig. 2a) optimum than the YO one (Yarrum optimum; Fig. 2b). The RMA regression of the observed D_{R} vs the Murray predicted D_{R} was not significantly different from the 1 : 1 line (slope = 1.08). The corresponding regression for observed N_{R} vs the Yarrum predicted N_{R} was significantly different from the 1 : 1 line (slope = 0.12), with measured N_{R} falling far short of the YO optimum. Greater agreement with ML was also supported by the much smaller mean predicted error for the ML optimum than the YO alternative (1.1 vs 10.9, respectively). For the individual species, Murray's law could not be rejected for 11 of the 13 species measured (Table 1). The comparison of the petiolule versus the third rank in A. pedatum also was consistent with ML. The two species that deviated from ML were tropical tree species (S. morototoni and D. retusa). The former had less conduit taper than predicted by ML, and the latter exhibited more taper than the optimum and was quite close to the areapreserving line (Fig. 2c).
Consistent with the tendency towards Murray's law, the data generally fell above the areapreserving line (Fig. 2c). Although the slope of the RMA regression in Fig. 2(c) was not significantly different from the areapreserving slope of 1, the yintercept was marginally greater than zero, which indicated that the lumen crosssectional area summed across the petiolule rank was greater than in the petiole for most species. The mean predicted deviation from the AP line was also greater than for the ML optimum (2.0 vs 1.0, respectively). This increase in area occurred despite the decline in vessel diameters from the petiole to petiolule ranks, because the number of vessels increased (Table 1). Areaincreasing conduit branching is predicted for Murray's law when N_{R} > 1.
Of most consequence to the plant is how much a given deviation from ML or YO anatomy costs in terms of lost conductivity per investment. Again, ML appears to define the more relevant optimum. Species averaged 97 ± 5% of the peak conductivity at ML (Fig. 3) vs 47 ± 26% of the YO value (not shown). Of the two species that deviated statistically from ML anatomy (S. morototoni and D. retusa), in only one of them did this result in a sizable deviation from ML conductance (Fig. 3; 82% of optimum). Interestingly, this species (D. retusa) fell close to the areapreserving condition (Fig. 2c). Excluding D. retusa, species averaged 98 ± 2% of their ML conductivity.
The general agreement with ML suggests that, within a species, D_{R} adjusts to a ‘preset’ value of N_{R} rather than vice versa as would be the case for the Yarrum optimum. In all species, N_{R} exceeded the often assumed value of 1 (West et al., 1997), ranging from 1.3 in the more distal comparison of A. pedatum to ∼6 in S. morototoni, and with a median of 2.4 ± 0.4 (Figs 2b, 4). An advantage of N_{R} > 1 is that, for a given leaf structure and vascular volume, the larger the N_{R} the greater the conducting efficiency at Murray's law (Fig. 1; McCulloh et al., 2003). Variation in N_{R} between species was at least in part related to differences in leaf structure: species with more leaflets per leaf tended to have greater N_{R} (Fig. 4).
Conduit number also scaled significantly with leaf and leaflet area within most species, and also across species (Fig. 5a). In order for N_{R} to be > 1 across petiole to petiolule ranks, the number of conduits must decrease less than the drop in leaf area moving from whole leaf to leaflet. Hence, the speciesspecific regressions of conduit number vs leaf and leaflet area supplied followed power functions with an exponent of < 1 (Fig. 5a; species log–log regressions, all slopes < 1), which is the value required for N_{R} = 1 (Appendix, Eqn A2). Exponents varied from −0.7 to 0.8 (Table 2), with the variation in part reflecting the different numbers of leaflets in the different species (Fig. 4). The negative exponents, for example, corresponded to the species with the fewest leaflets per leaf (A. candicans and S. riparium). There was also significant variation between species in the number of conduits for a given leaf area, as indicated by significant differences in the intercepts of the log–log relationships (Table 2). Pooling across all species indicated a significant increase in conduit number with area supplied with an exponent of 0.53. The low R^{2} of this pooled regression (0.35) reflects the variation in slopes and intercepts between species: four of the 13 species had slopes significantly different from the pooled regression, and nine species had different intercepts (Table 2).
Table 2. The slope, R^{2} values and yintercepts from the reduced major axis regressions shown in Fig. 5Species  Vessel number vs leaf area  Vessel diameter vs leaf area  Theoretical conductivity vs leaf area 

Slope (± CI)^{a}  R^{2}  yintercept  Slope (± CI)  R^{2}  yintercept  Slope (± CI)  R^{2}  yintercept 


Arrabidaea candicans  −0.68 (0.38)^{b}  0.54  3.63  0.36 (0.13)  0.81  0.53  −0.85 (0.68)  0.82  0.43 
Dalbergia retusa  0.53 (0.08)  0.97  1.37  0.26 (0.02)  0.99  0.67  1.57 (0.11)  0.99  −4.13 
Paullinia pterocarpa  0.62 (0.12)  0.94  0.87  0.14 (0.06)  0.71  0.90  1.14 (0.21)  0.97  −3.64 
Schefflera morototoni  0.28 (0.05)  0.96  2.04  0.19 (0.02)  0.98  1.07  1.09 (0.09)  0.99  −2.05 
Serjania cornigera  0.21 (0.09)  0.72  1.90  0.28 (0.05)  0.95  0.76  1.29 (0.18)  0.97  −3.20 
Stizophyllum riparium  −0.54 (0.42)  0.11  3.38  0.43 (0.17)  0.76  0.48  1.41 (0.42)  0.87  −3.36 
Clematis armandii  0.82 (0.48)  0.49  0.68  0.30 (0.11)  0.80  0.63  1.73 (0.47)  0.89  −4.50 
Daucus carota  0.62 (0.23)  0.79  1.30  0.22 (0.13)  0.52  0.88  1.42 (0.63)  0.71  −3.28 
Fragaria ¥ananassa  0.39 (0.29)  0.16  2.37  0.24 (0.09)  0.80  0.52  1.08 (0.32)  0.87  −3.40 
Rubus discolor  0.32 (0.16)  0.64  2.04  0.29 (0.09)  0.85  0.69  1.35 (0.27)  0.94  −3.22 
Sambucus caerulea  0.35 (0.16)  0.70  1.50  0.33 (0.04)  0.98  0.73  1.63 (0.26)  0.96  −3.73 
Adiantum pedatum  0.47 (0.24)  0.74  1.26  0.29 (0.23)  0.36  0.55  1.60 (1.07)  0.55  −4.72 
Pteridium aquilinum  0.70 (0.13)  0.95  0.84  0.20 (0.13)  0.40  0.88  1.43 (0.58)  0.76  −3.68 
Pooled  0.53 (0.08)  0.35  1.38  0.31 (0.03)  0.71  0.62  1.50 (1.12)  0.82  −3.79 
Average vessel diameter also scaled significantly with leaf area supplied both within and between species (Fig. 5b). Speciesspecific scaling exponents ranged from 0.1 to 0.4, which corresponds well with the 0.1 to 0.6 range predicted for Murray's law from the N_{R} scaling in Fig. 5(a) (Appendix, Eqn A7). The pooled data exhibited an exponent of 0.31 for diameter vs leaf area scaling, deviating slightly from the 0.16 value for perfect ML compliance across species (Fig. 5b, dotted ML line; Appendix, Eqn A7). The R^{2} of 0.71 for the pooled regression suggests greater convergence of intra and interspecific scaling for conduit diameters than for conduit numbers (Fig. 5a). Only three of the 13 species had slopes different from the pooled value, and three species had different intercepts (Table 2).
The greatest convergence between intra and interspecific scaling was in the calculated conductivity vs leaf area (Fig. 5c). Petiole or petiolule theoretical conductivity scaled with leaf area supplied to the 1.50 power across species (Fig. 5c; Table 2). The relatively high R^{2} of 0.82 for the pooled regression reflected the fact that only one of 13 species deviated from the pooled regression for slope and one for intercept. The pooled slope of 1.50 is greater than the exponent of 1 for conductivity increasing in direct proportion to leaf area (Fig. 5c; dotted ‘KL’ line for constant leafspecific conductivity). An increase in leafspecific conductivity with leaf area was consistent with the general compliance with ML, which predicts a conductivity vs leaf area exponent of 1.16 (Fig. 5c, ML line; Appendix Eqn A10). Computing conductivities on the basis of the actual conduit distributions rather than the average conduit diameter did not change the indication that conductivity increases disproportionately with leaf area (R^{2} = 0.83, slope = 1.23; not shown).
Discussion
 Top of page
 Summary
 Introduction
 Materials and Methods
 Results
 Discussion
 Acknowledgements
 References
 Appendix: Relations between scaling exponents in
Compound leaves from a wide range of species complied more closely with Murray's law (ML) than with the Yarrum optimum (YO; Fig. 2a vs 2b). Only two species deviated statistically from ML (Table 1), but even the greatest deviant (D. retusa) was within 18% of the broad optimal ML conductivity (Fig. 3; Sherman et al., 1989). These results suggest that vessel diameter ratio (D_{R}) ‘tunes’ to a constrained number ratio (N_{R}) and not vice versa. This sequence makes sense for several reasons. First, the number of conduits (hence N_{R}) is determined much earlier in development than conduit diameter (hence D_{R}). Yarrum's N_{R} also depends on the ratio of petioluletopetiole length (L_{R}), which like D_{R} is determined fairly late in development. Secondly, although plants cannot achieve the theoretical N_{R} max of a maximally efficient singletubed vascular network, they may have evolved to achieve the ‘next best’N_{R} within speciesspecific constraints of leaflet number (Fig. 4) and considerations of transport safety and redundancy. Finally, achieving the Yarrum optimum requires an even more extreme areaincreasing branching of conduits than Murray's law (Fig. 1), which may not be possible within the confines of petiole/petiolule architecture. Although it is known that vessel diameters increase basipetally as the vascular auxin concentration declines (Aloni & Zimmermann, 1983), how the plant develops the particular ML diameter ratio is unknown.
Conduits within woody stems tend to be closer to area preserving than the ML optimum, suggesting that the need to avoid topheavy networks constrains the hydraulic efficiency of trees (McCulloh et al., 2004; Atala & Lusk, 2008). It is worth noting that this area preservation is not da Vinci's rule, which refers to preservation of the total crosssectional area of branches, not their internal conducting tubes (Horn, 2000). Interestingly, the compound leaf that deviated most significantly from Murray's law (leaves of D. retusa; Fig. 2a) was fairly close to the areapreserving line (Fig. 2c). This species had 12–14 leaflets arranged pinnately on a long, rigid axis and perhaps relied more than other species on its xylem conduits for mechanical support. This reliance may have prevented areaincreasing branching of vessels which otherwise would have been more hydraulically efficient.
Givnish (1978) suggested that a benefit of compound leaves was as ‘throwaway branches’ that could achieve the same area as simple leaves for a smaller investment in woody biomass. Here, we propose that, by following Murray's law more closely than stems, compound leaves are also more hydraulically efficient than the equivalent stem structure. The efficiency is achieved by a greater uncoupling of mechanical support from hydraulic supply, which allows the conduit network to be areaincreasing as required for the most efficient Murray law networks.
Intraspecific scaling of the petiole–petiolule network with leaf area did not carry over precisely to a single universal interspecific scaling (Fig. 5). The most variation between species was in how the number of conduits scaled with leaf area (Fig. 5a). However, consistent with N_{R} being > 1, all intraspecific exponents were < 1; as was the pooled exponent (Table 2). Diameter scaled more consistently across species with leaf area, but still with significant interspecific variation (Fig. 5b). This result is consistent with the strong degree of scaling between the hydraulically weighted vessel diameter and leaf area observed by Coomes et al. (2008) across multiple vein orders in leaves of various oak species. The variation in diameter scaling observed in our data partially compensated for variation in conduit number, because theoretical conductivity (calculated from diameter and number) exhibited a fairly strong interspecific relationship with leaf area (Fig. 5c). Actual conductivity tends to be proportional to the theoretical value (Sperry et al., 2006; Choat et al., 2008). These data imply that leafspecific conductivity of petioles increases with leaf area, a trend consistent with following Murray's law. Experimental data on hydraulic conductivities measured in petioles of simple leaves are consistent with this trend (Sack et al., 2003).
Unlike petiole conductivity, petiole conductance (= conductivity/length) may be relatively invariant with leaf size because petioles tend to be longer in larger leaves (Niklas, 1994; Sack et al., 2003). A recent survey found that petiole length scaled with leaf area to approximately the 0.5 power (Arcand et al., 2008). Combination of this with the 1.5 scaling of conductivity (Fig. 5c) would predict that petiole conductance scales isometrically with leaf area as does lamina conductance (Sack et al., 2003). Variation in conductivity or conductance by leaf area scaling may be related in part to light environment and the prevailing evaporative gradient, with sunlit leaves or leaves in drier air developing greater conductivity for a given leaf area than shaded leaves or leaves in humid air (Sack et al., 2003).
The petioles and petiolules of compound leaves are discrete branching ranks that provide an opportunity to examine compliance with two distinct optima that both maximize hydraulic conductance per investment in vascular tissue volume. The Yarrum optimum does not seem to represent an achievable optimum, suggesting that N_{R} is more constrained than D_{R} in the evolution and development of the leaf vasculature. Most species tracked the alternative Murray law optimum. Convergence on a hydraulic optimum suggests the importance of hydraulic efficiency for maximizing plant fitness. Deviations from the ML optimum in some leaves and in most stems tend to approach the alternative areapreserving behavior, a more adaptive alternative when the xylem conduits are called upon to supply mechanical support in addition to water.
Appendix: Relations between scaling exponents in Fig. 5
 Top of page
 Summary
 Introduction
 Materials and Methods
 Results
 Discussion
 Acknowledgements
 References
 Appendix: Relations between scaling exponents in
Figure 5(a) shows that the number of conduits (N) is proportional to leaf or leaflet area (A) to the ‘x’ power:
Accordingly, the number of conduits per petiolule/number of conduits per petiole ≈ (A_{i}/A_{L})^{x}, where A_{i} is the average leaflet area and A_{L} is the whole leaf area. The approximation is because mean leaflet area is used and the proportionality in Eqn A1 is estimated from a regression. Multiplying (A_{i}/A_{L})^{x }by the number of petiolules (≈ A_{L}/A_{i}) yields
 N_{R} ≈ (A_{i}/A_{L})^{x} (A_{L}/A_{i}) (Eqn A2.)
Using A_{R} = A_{i}/A_{L},
 N_{R} ≈ A_{R}^{(x–1)} (Eqn A3.)
Because A_{R} < 1, it is clear that for N_{R} > 1 as observed, x must be < 1.
Fig. 5(b) shows that the average diameter of conduits (D) is proportional to leaf area to the ‘y’ power:
 (Eqn A5)
From Eqn 6, at Murray's law. Equating A3 and A5 at Murray's law gives:
 (Eqn A6)
For x = 0.53, the pooled regression exponent from Fig. 5(a), y = 0.16 for Murray's law (ML line in Fig. 5b).
Figure 5(c) shows that conductivity (K) scales with A to an exponent ‘z’:
Assuming that KαND^{4}, then from A1 and A4:
which means z = x + 4y. Plugging in y from A7 gives z as a function of x for Murray's law:
 z = 1.33 − 0.33x (Eqn A10)