Tree Physiology 28, 1–10
© 2008 Heron Publishing—Victoria, Canada
Applying a universal scaling model to vascular allometry in a
single-stemmed, monopodially branching deciduous tree (Attim’s
model)
PEKKA NYGREN1,2 and STEPHEN G. PALLARDY3
1
Department of Forest Ecology, P.O. Box 27, 00014 University of Helsinki, Finland
2
Corresponding author ([email protected])
3
Department of Forestry, 203 ABNR Building, University of Missouri, Columbia, MO 65211, USA
Received November 8, 2007; accepted May 11, 2007; published online October 15, 2007
Summary West, Brown and Enquist (1999a) modeled vascular plants as a continuously branching hierarchical network
of connected links (basic structural units) that ends in a terminal unit, the leaf petiole, at the highest link order (WBE
model). We applied the WBE model to study architecture and
scaling between links of the water transport system from lateral
roots to leafy lateral branches and petioles in Populus deltoides
Bartr. ex Marsh. trees growing in an agroforestry system
(open-grown trees) and in a dense plantation (stand-grown
trees). The architecture of P. deltoides violates two WBE
model assumptions: (1) the radii of links formed in a branching
point are unequal; and (2) there is no terminal unit situated at
the end of a hierarchical network, rather, petioles are situated at
any link order greater than 1. Link cross sections were taken at
various link orders and morphological levels in roots and
shoots of open-grown trees and shoots of stand-grown trees.
Scaling of link radii was area-preserving. From roots to
branches, vessel diameters were scaled with link order in accordance with a 1/6-power, as predicted by the WBE model indicating general vessel tapering. However, analysis of the data
at the morphological level showed that vessel radius decreased
intermittently with morphological level rather than continuously between successive link orders. Estimation of total water
conductive area in a link is based on conducting area and petiole radius in the WBE model. The estimation failed in P. deltoides, probably because petioles are not a terminal unit. Biomass of stand-grown trees scaled with stem basal radius according to the 3/8-power predicted by the WBE model. Thus,
the WBE model adequately described vascular allometry and
biomass at the whole-tree level in P. deltoides despite violation
of Assumption 1, but failed in predictions where the leaf petiole
was used as a terminal unit.
Keywords: fractal network, hydraulic architecture, pipe
model, Populus deltoides, quarter-power scaling, tree architecture, WBE model.
Introduction
Trees cover a wide range of taxonomic and structural diversity,
and a tree may span more than 12 orders of magnitude in size
during its development from seedling to maturity. However, all
trees share essentially the same anatomical and physiological
design. Hallé and Oldeman (1970) were the first to show that
the vast diversity of tree forms may be reduced to a few architectural models. Hallé et al. (1978) described 23 architectural
tree models, but more importantly, their work indicated that all
trees are composed of a few basic, repeating structural elements. This idea has been used in developing functional–structural tree models (Room et al. 1994, Godin et al. 1999,
Sievänen et al. 2000).
Leonardo da Vinci was the first to observe that tree branching follows the same rules as branching of a water course; in
his attempt to draw a perfect tree, he postulated that the sum of
the thickness of all branches is equal to the thickness of the
subtending stem before the branching point (Zimmermann
1983). Shinozaki et al. (1964) formalized the pipe model theory by proposing that plant stems are “an assemblage of unit
pipes each supporting a unit amount of photosynthetic organs.” Pipe model theory has been used with varying success
for modeling tree structure and dry matter allocation (reviewed by Zimmermann 1983, Sievänen et al. 2000). The pipe
model works best in relatively fast-growing tree species. However, the pipe model relationships seem to be species- and
site-dependent, and the hydraulic structures of the stem,
branches and roots seem to differ in many species, leading to
different pipe model scaling factors in different organs (Zimmermann 1983).
Because of these difficulties, the application of fractal geometry for modeling tree structure has received increasing attention (Zeide and Pfeifer 1991, van Noordwijk et al. 1994,
Berntson 1996, West et al. 1997, 1999a). Sapwood branching
according to the pipe model theory is a special case of a more
general fractal scaling pattern of biological transport systems
2
NYGREN AND PALLARDY
(West et al. 1997). Fractal geometry was developed for describing the natural forms that cannot be accommodated by
Euclidean geometry (Mandelbrot 1983). Fractal objects that
follow the self-similarity principle, i.e., maintain similar form
over a range of scales (Mandelbrot 1983), are of special interest for modeling tree structure. Fractal geometry applied to a
tree predicts that the shoots or roots follow the same bifurcation pattern from stem or proximal roots to the leaf petioles or
smallest transport roots. However, real trees are usually only
partially self-similar because link radii and length scale by different factors (West et al. 1999a, Salas et al. 2004) leading to
non-similar variation in link volume.
Vascular allometry has been proposed as a universal scaling
principle for the size of trees and other organisms that rely on
an internal transport network (West et al. 1997, 1999a, 1999b,
Enquist 2002, Enquist and Niklas 2002; hereafter WBE
model). According to the WBE model, a tree is a continuously
branching hierarchical network of connected links (Figure 1).
Link length and diameter depend on link order, but scaling factors between subsequent links are invariant within a whole
tree. The link order is assigned following a developmental sequence (cf. Berntson 1996), i.e., starting from the root collar
upward (shoot) or downward (root). The stem is a link of Order 0. The approach is also called the quarter-power scaling
law because scaling exponents are often multiples of 1/4
(West et al. 1997, 1999b). The links of the WBE model or
other fractal tree models are not equivalent to growth units in
the architectural classification of Hallé et al. (1978).
The WBE model is based on four assumptions: (1) wholeplant architecture is volume-filling; (2) within a species and
during ontogenetic development of an individual, the physical
dimensions of the leaf petiole are approximately invariant; (3)
biomechanical constraints are uniform; and (4) hydrodynamic
resistance throughout the vascular network is minimized. Further, it is assumed that all xylem vessels are aligned in parallel
and run continuously from root tip to leaf and are of equal
length, and xylem vessels have a constant radius within a link
but radii may vary between links allowing for vessel tapering
(Enquist 2002). The strength of the model is a good fit of the
quarter-power scaling law to large datasets (Enquist and
Niklas 2002, Enquist 2003). However, the usefulness of the
model for analyzing the structure of individual trees is not
self-evident (Meinzer et al. 2005).
Implicit assumptions of the WBE model are that a vascular
network is divided at each branching point into n branches of
equal radius (West et al. 1997; Figure 1) and that terminal
units are all situated at a link of Order N (West et al. 1999a).
This topology produces only two architectural models, Leeuwenberg’s and Schoute’s models, out of the 23 architectural
tree models described by Hallé et al. (1978). In all other models, the link order of leaf petioles—the terminal unit of the
WBE model—may vary (Figure 1). Eastern cottonwood (Populus deltoides Bartr. ex Marsh.) is a single-stemmed, monopodially branching tree that grows continuously during a growing season. It represents Attim’s model according to the classification of Hallé et al. (1978). The architecture of P. deltoides
does not fit the above stated assumptions of the WBE model,
i.e., link radii after a branching point are not equal and may
differ substantially (Nygren et al. 2004), and terminal units,
leaf petioles, are not at one terminal level of Order N but may
have any link order greater than 1 (Figure 1). Thus, P. deltoides provided an opportunity to test the predictions of the
WBE model at the individual species level, a context quite distinct from that involving large, multi-species datasets in which
important species-dependent attributes may be obscured by
averaging effects.
The objective of our study was to apply principles of vascular allometry to describe scaling and architectural features of
P. deltoides and compare WBE model predictions with field
data at the individual-tree level.
The WBE model
A tree is described as a continuously branching hierarchical
network originating from the main stem, assigned Order 0, to
the petioles of Order N (West et al. 1997, 1999a, Enquist
2002). Tree architecture is characterized by three parameters
that relate subsequent link radii (r)
a
–
rk + 1
=n 2
rk
(1)
vessel radii (v)
b
–
vk +1
=n 2
vk
Figure 1. Scheme of tree branching and link orders as assumed in the
WBE model (West et al. 1997; left) and branching pattern of Populus
deltoides trees (right). The basal link (Order 0) has the same diameter
in both cases. According to the example shown, leaf petioles are of
Order 4 in the WBE model because they are attached only at the highest branches, but may have any order from 2 to 5 in P. deltoides.
(2)
and link lengths (l)
1
–
lk + 1
=n 3
lk
TREE PHYSIOLOGY VOLUME 28, 2008
(3)
VASCULAR ALLOMETRY IN A MONOPODIAL TREE
where n is the number of links formed in a branching point, k is
the link order, and a and b are scaling parameters. The parameter values are assumed to be independent of k. Equation 1 formalizes the implicit assumption that link radii after a branching point are equal. The model has been developed for plant
shoots. We extended the model to include the root system by
assigning Order –1 to roots attached to the root collar, and
more negative numbers to each root branching generation till
the root tip. This formalism means that petiole level N does not
give the total number of branching generations as in the original WBE model, but the total number of branching generations
is given by subtracting the lowest root order from N.
The WBE model produces testable predictions on the scaling of the vascular system and plant biomass. First, it predicts
that water conductive area, i.e., the sum of cross-sectional areas of all vessels, at any level k (Ac,k) is related to the conductive area in a petiole (Ac,N ):
Ac, k = Ac, N
 rk 
 
 rN 
2( 1 + b )
field floods occasionally in spring and early summer. The soil
is a Moniteau silt loam (fine-silty, mixed, superactive, mesic
Typic Endoaqualf) that is fertile, moderately well-drained and
permeable (Pallardy et al. 2003).
One plantation was an alley cropping experiment (hereafter
“open-grown trees”). The plantation was established with
1-year-old cuttings at a 6 × 18 m spacing in April 2001 in association with white clover (Trifolium repens L.). All trees were
of a Midwestern USA industry clone. No within-row canopy
closure had occurred before the measurements in summer
2003, thus, the alley cropping was considered to be an opengrowth plantation. Mean tree height was 4.9 m in fall 2003, after three growing seasons. The second plantation was a dense
stand established with 20-cm long cuttings in May 1999. The
stand was planted at a 1 × 1 m spacing with three P. deltoides
clones and a poplar hybrid (Pallardy et al. 2003). Only
P. deltoides clones were sampled. Mean height of P. deltoides
was 9.3 m in January 2004, after five growing seasons.
a
(4)
where rk is the link radius at level k and rN is petiole radius
(West et al. 1999a). Second, it is argued that the hydrodynamic
resistance of the plant vascular system is minimized when parameter b = 1/6, or when there are only a few (< 30) branching
generations (West et al. 1999a):
b=
3
1
1
+
6 2 N ln n
(5)
However, it has recently been shown that the parameter b value
of 1/6 does not minimize hydrodynamic resistance (Mäkelä
and Valentine 2006).
According to the WBE model, at the whole-tree level,
height (H ) scales with stem basal radius (r0):
Vascular measurements
Three open-grown trees and three stand-grown trees were
sampled for vascular allometry in July and August 2003. In the
stand, one tree per clone was sampled. The vascular samples
were taken at various branching orders as defined by West et
al. (1997; Figure 1) and different morphological levels (Figure 2). Because the main experiment in the stand did not permit root excavations at the time of sampling, root architecture
was sampled only in open-grown trees. We analyzed 127 link
cross sections in open-grown trees and 76 in stand-grown
trees.
Samples were collected in the field early in the morning,
placed in plastic bags and transported without delay to the laboratory, where they were stored at 5 °C until measured. For
petiole measurements, a whole leaf was placed in a plastic bag
and a cross section was cut from the middle of the petiole with
2
H ∝ r0 3
(6)
and woody biomass (M ) scales with r0:
r0 ∝ M
3a
2( a + 3 )
(7)
Theoretically, the value of the exponent of biomass in Equation 7 should be about 3/8 (Enquist 2002). This is the case
when a = 1 or branching is area preserving (West et al. 1999a).
Materials and methods
Field data
Field data were measured in two adjacent Populus deltoides
plantations on the Missouri River flood plain at the University
of Missouri’s Horticulture and Agroforestry Research Center
in New Franklin, MO (39°01′ Ν, 92°46′ W, 197 m a.s.l.). The
Figure 2. Morphological levels identified in Populus deltoides trees:
stem (thick line above ground); leafless main branches attached to
stem (thin solid lines above ground); leafy terminal branches (dashed
lines above ground), which are attached to main branches; main roots,
which are attached to the root collar (medium solid lines below
ground); and lateral roots, which are attached to main roots (dotted
lines below ground). The thickest link after each root junction was
considered an extension of the main root.
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4
NYGREN AND PALLARDY
a razor blade just before measurements were made with the aid
of a stereomicroscope. Cross sections from branches and roots
were obtained with a razor blade or knife, depending on organ
diameter. Stem samples were taken with an increment borer to
the center of the stem. The upper side of the core was marked
before removing the sample from the borer. Stem diameter at
the sampling point was measured with a caliper to 0.1 mm.
Petioles and small green branches were measured on the day
of sampling. Woody samples were measured within 2 days of
sampling. Samples were measured with the aid of a stereomicroscope at the Dendrochronology Laboratory, University
of Missouri. Each sample was placed on a moving stage. The
stage movements were monitored by a displacement transducer to 0.01 mm. The sample surface was cleaned with a razor blade when necessary.
The tissues in cross sections of woody samples were divided
into bark, phloem and cambium—which were pooled together—sapwood and pith. No heartwood was apparent in any
sample. Width of each tissue was measured in two perpendicular directions. The area of each section was calculated based
on the mean width of the tissue segment assuming that the total
cross section was round; little variation between the measurement directions was observed, and the form was therefore considered circular. The same sections were identified from stem
cores, and their widths were measured. The upper side of the
stem core was cleaned with a razor blade before measurements
were made. Cross sections of petioles were ellipsoidal, and the
cross-sectional area of a petiole was determined as an ellipse
by measuring lengths of long and short axes. Xylem vessels
were clustered in petioles rather than diffusely distributed in a
clearly defined xylem as in woody samples.
Diameters of 10 vessels were measured for each sample,
and their mean was used in subsequent analyses. Petiole vessel
density was determined by counting the number of all vessels
in the petiole cross section and dividing the vessel number by
petiole cross-sectional area. Sapwood of P. deltoides and other
poplars and cottonwoods is diffuse-porous (Brown et al. 1949,
Zimmermann 1978), i.e., vessels are not radially distributed in
the sapwood. Vessel diameter may vary between spring and
summer wood (Brown et al. 1949) and because of environmental conditions (Schume et al. 2004) but no systematic pattern can be observed between annual growth rings of an individual tree (Mátyás and Peszlen 1997). Therefore, vessel density of woody samples was determined based on four count areas, which formed a cross pattern through a sapwood cross
section with the central pith excluded. The length of each area
was the length from the sapwood’s outer edge to the pith and
width was twice the mean vessel diameter in the sample. Only
vessels completely within this area were counted. Vessel density was calculated by dividing the number of vessels by the
count area. Total water conductive area was determined as the
product of vessel cross-sectional area, vessel density and sapwood area of a cross section.
Biomass measurement
Biomass samples were derived from stand-grown trees in replicate plots of 70 trees (10 rows × 7 columns at 1 × 1 m spac-
ing) per clone in a randomized complete block design. During
each year of growth of the 5-year-old plantation, several trees
were harvested across a range of tree sizes at the end of each
growing season. Sampling was designed to harvest trees buffered by border trees. Before harvest, tree height and basal diameter (about 10 cm from the ground) were measured. Stem
and branch material was bulked, placed in convection ovens
and dried at 70 °C to constant mass. Root excavation was also
conducted. Soil and roots were removed in a 1 m2 square centered on the tree stem to about 1 m depth, resulting in the excavation of one cubic meter of soil. This method allowed for a
close estimate of tree root biomass, with roots of non-sample
trees growing into the excavated volume compensating for
roots of the sample trees growing outside the cubic meter zone
(Scarascia-Mugnozza et al. 1997). Roots growing below 1 m
in depth were limited to no more than three or four brace roots
that terminated at depths of less than 25 cm beyond the 1-m excavation limit. As soil was removed from the root volume it
was placed in plastic containers and separated from roots by
hand. Once roots arrived at the laboratory they were washed
free of soil using a 1 mm sieve, placed in an oven and dried at
70 °C to constant mass. In total, 48 P. deltoides trees were
harvested in this manner over 5 years.
Data analyses
Curves were fit either by linear regression or by nonlinear regression with the Marquardt iteration algorithm. Because vessels were not sampled at all branching orders, parameter b was
estimated from Equation 2 in the form:
–
vk + q = vk n
qb
2
(8)
where q is the number of branching orders between two subsequent vessel diameter measurements. We used n = 2 in all calculations. The value of parameter b was determined by nonlinear curve fitting.
Equation 6 was written as:
H = β H r0
αH
(9)
Equation 9 was resolved by two alternative methods: (1) by
setting the predicted value H = 2/3 and resolving H by linear
regression without intercept, and (2) by resolving both parameter values by nonlinear curve fitting. To estimate the scaling
exponent between r0 and M, Equation 7 was written as:
1
M = ( β M r0 ) α M
(10)
Two values of M were tested: M = 3a/(2(a+3)) with the value
of parameter a determined from data and M = 3/8 as predicted
by the WBE model.
The proportion of explained variance (r 2 ) of the equations
was computed (Mayer and Butler 1993) as:
TREE PHYSIOLOGY VOLUME 28, 2008
VASCULAR ALLOMETRY IN A MONOPODIAL TREE
∑( y i – y$i )
=1–
2
∑( y i – y )
5
2
r
2
(11)
where yi is the ith observation of the dependent variable, y$ i is
the estimate for the ith observation of the dependent variable,
and y is the mean of the observations of the dependent variable. The numerator is the uncorrected sum of squares of
model residuals, and the denominator is the corrected sum of
squares of observations. In the case of nonlinear curve fitting,
the upper limit of r 2 is 1, as in linear regression, but the lower
limit is negative infinity because a curve of fixed form is fitted
to the data (Mayer and Butler 1993).
Results
Vessel radius
Parameter b, which describes scaling of individual vessels,
was estimated by nonlinear curve fitting of Equation 8 to vessel radius data at different link orders. Petioles were excluded
from fitting. Parameter b could be solved for open-grown trees
(Figure 3), but fitting to data of stand-grown trees failed; the
negative r 2 value indicates that the curve and data points were
independent of each other (Mayer and Butler 1993). The value
of b for open-grown trees was 0.223, which is close to the
value of 0.192 predicted by Equation 5 for 29 branching orders
(from Order –8 for lowest lateral roots to Order 21 for the
highest leafy branches). An estimate of vessel radius was calculated recursively with Equation 2 and setting the vessel radii
of the lowest order, v–8 in open-grown trees and 0 in standgrown trees, to the observed mean values of 78.5 µm and
33.2 µm, respectively. The estimate satisfactorily fit the opengrown tree data at the whole-tree level but failed to fit the
stand-grown tree data (Figure 3).
Despite the satisfactory whole-tree fit of Equation 2 to the
open-grown tree data, vessel radius was quite invariant at each
shoot morphological level, whereas a slight, although not statistically significant, pattern of decreasing vessel radius with
increasing link order was observed in roots (Figure 3). Linear
regressions were fit to describe the vessel radius as a function
of link order. The intercepts of regression lines differed significantly from zero, but the slopes did not differ significantly
from zero except for leafy branches in open-grown trees, indicating that vessel radius was almost invariant at each morphological level (Figure 3). Analysis of variance revealed that differences in vessel radius between morphological levels were
significant (Table 1). Variation (indicated by SE) within each
morphological level was low except for lateral roots. Vessel radii in main and leafy branches were significantly larger in
open-grown trees than in stand-grown trees, whereas no significant differences were observed in stems and petioles.
Attempts to estimate parameter a from data on link radii at
different link orders failed, resulting in negative r 2 values for
both open- and stand-grown trees, probably because of large
variation in link radii at all branching orders.
Figure 3. Vessel radius as a function of link order in Populus deltoides
trees in (a) an open agroforestry plantation and (b) a dense stand. Negative link orders refer to roots, and positive link orders refer to shoots.
The estimate was calculated recursively according to Equation 2 with
b = 0.223, which was the best-fit value for open-grown trees. The vessel radius was set as 78.5 µm at the link of Order –8 (lateral root) in
open-grown trees and 33.2 µm at the link of Order 0 (root collar) in
stand-grown trees, according to measurements. The estimate fitted
satisfactorily to the open-grown trees data (root mean square error,
RMSE = 14.23, r 2 = 0.59, n = 109) but did not fit to stand-grown tree
data (RMSE = 8.27, r 2 = –0.26, n = 67). The dashed lines indicate the
linear regressions between link order and vessel radius within each
morphological level; slope was significantly different from zero only
for leafy branches in open-grown trees, but all intercepts were significantly different from zero.
Water conductive area
Vessel radii of leaf petioles varied relatively little between
open- and stand-grown trees (Table 1). The same was true for
petiole radii, r N: the mean ± SE was 1.740 ± 0.1001 and
1.573 ± 0.0670 mm in open- and stand-grown trees, respectively. The water conductive area of petioles, Av,N, also varied
little, being 0.070 ± 0.009 and 0.052 ± 0.008 mm2 in open- and
stand-grown trees, respectively. Water conductive area in
woody parts of a tree was estimated as a function of link radius
applying Equation 4 by setting rN and Av,N to these measured
values. A b value of 0.223 was used for both open- and
stand-grown trees. Parameter a resolved by nonlinear curve
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6
NYGREN AND PALLARDY
Table 1. Mean vessel radius (± SE) at different morphological levels
of Populus deltoides trees in an open agroforestry plantation (opengrown trees) and a dense stand (stand-grown trees). Figures in parenthesis are numbers of observations in each case. Means within a column followed by the same letter do not differ significantly (Duncan’s
Multiple Range Test at 5%).
Morphological level
Lateral root
Main root
Stem
Main branch1
Leafy branch1
Leaf petiole
1
Vessel radius (µm)
Open-grown trees
Stand-grown trees
57.3 ± 3.02 a (30)
64.1 ± 4.15 a (21)
36.6 ± 1.49 b (10)
22.8 ± 0.72 c (15)
16.7 ± 0.92 cd (38)
10.6 ± 0.52 d (23)
–
–
32.9 ± 1.14 a (9)
15.1 ± 0.94 b (25)
12.6 ± 0.60 bc (33)
10.9 ± 0.68 c (19)
Vessel radius differs significantly between sites in these morphological levels.
related to link radius but not to petiole traits. Further, the value
of a approaching unity (from Equation 13) suggests an areapreserving branching pattern in P. deltoides.
Conductive area to total sapwood area ratio decreased significantly from roots to stem to branches in open-grown trees
(Table 2). In stand-grown trees, leafy branches had a higher ratio than main branches in contrast to open-grown trees. The ratios were significantly higher in open-grown trees than in
stand-grown trees at all shoot morphological levels. The ratio
was not computed for petioles because their anatomy differed
from woody tissue.
Because conductive area varied little in petioles, the relationship of conductive area in leafy branches to number of
leaves on the branch was analyzed (Figure 5). Both linear and
quadratic regressions were fitted to data accounting for the
conductive area as a function of the number of leaves above the
sampling point in a branch. The quadratic regression without
fitting was 0.829 for open-grown trees (r 2 = 0.96) and 0.853
for stand-grown trees (r 2 = 0.93). Although the general fit of
Equation 4 to the data was good, the original WBE model produced underestimates at low link radii (Figure 4).
Because petioles are not situated at a single terminal link order in Populus deltoides, we tested whether the water conductive area is related to link radius:
Ac, k ∝ rk
(12)
Equation 12 was resolved as a linear regression between
log(Ac,k ) and log(rk ). Fitting the linear regression to the data
eliminated underestimation (Figure 4) and slightly improved
the r 2 values (0.98 for open-grown trees and 0.94 for standgrown trees). If Equation 4 holds, the slope of the log–log regression, r , is:
αr =
2(1 + b )
a
(13)
Setting b = 0.223 gives values of a close to one (1.011 and
1.028 for open- and stand-grown trees, respectively). Further,
the intercept term of the log–log regression, r , is related to
Equation 4 as:
10 β r = cAc, N
(14)
where
1
c= 
 rN 
αr
(15)
Applying the estimated r value (Figure 4) gives cAc,N =
0.0183 and cAc,N = 0.0177 for stand-grown and open-grown
trees, respectively, whereas 10 βr = 0.0961 for open-grown
trees and 0.0532 for stand-grown trees. These analyses indicate that, in P. deltoides, water conductive area within a link is
Figure 4. Water conductive area as a function of vessel radius in
Populus deltoides trees in (a) an open agroforestry plantation and (b) a
dense stand. The WBE model estimate was calculated according to
Equation 4 with a = 0.829 (r 2 = 0.96) and 0.853 (r 2 = 0.93) for openand stand-grown trees, respectively. The statistical estimate is a linear
regression between link radius and water conductive area after
log–log transformation. The intercept term ( r ) and slope ( r ) were
–1.017 and 2.419 (r 2 = 0.98) for open grown trees, respectively, and
–1.275 and 2.380 (r 2 = 0.94) for stand-grown trees, respectively.
TREE PHYSIOLOGY VOLUME 28, 2008
VASCULAR ALLOMETRY IN A MONOPODIAL TREE
Table 2. Mean conductive area to total sapwood area ratio (± SE) at
different morphological levels of Populus deltoides trees in an open
agroforestry plantation (open-grown trees) and a dense stand (standgrown trees). Figures in parenthesis are numbers of observations in
each case. Means within a column followed by the same letter do not
differ significantly (Duncan’s Multiple Range Test at 5%).
Morphological level Conductive to sapwood area ratio
Lateral root
Main root
Stem1
Main branch1
Leafy branch1
1
Open-grown trees
Stand-grown trees
0.195 ± 0.013 a (30)
0.168 ± 0.011 a (21)
0.114 ± 0.007 b (10)
0.096 ± 0.003 bc (15)
0.077 ± 0.005 c (38)
–
–
0.092 ± 0.008 a (9)
0.043 ± 0.004 b (25)
0.062 ± 0.004 c (33)
The ratio differs significantly between sites in these morphological
levels.
an intercept fit the data for open-grown trees better than did the
linear regression (r 2 = 0.93 for quadratic and r 2 = 0.88 for linear). The fit of both regressions to stand-grown tree data was
poor (r 2 = 0.44 for quadratic and r 2 = 0.38 for linear).
Figure 5. Water conductive area in leafy branches as a function of
number of leaves above the measuring point in Populus deltoides trees
in (a) an open agroforestry plantation and (b) a dense stand (b). Two
statistical estimates, linear and quadratic regression without intercept,
are shown with data points.
7
Tree height and biomass
Whole-tree-level predictions of the relationship between stem
basal radius and tree height or biomass were tested for standgrown trees only. The relationship between stem basal radius
and tree height (Equation 9) predicted by the WBE model
overestimated height in small trees and underestimated it in
large trees (Figure 6). Nonlinear best fit for both the normalization constant H and exponent H of Equation 9 resulted in
2
H = 2.152 and H = 0.985 (r = 0.93). Thus, the relationship
between stem basal radius and tree height was essentially linear as opposed to the predicted relationship with an exponent
of 2/3 (Enquist 2002).
Equation 10 fit well to the relationship between stem basal
radius and whole-tree woody biomass with both the predicted
exponent value of 3/8 and with the value proposed in Equation 7 when a = 1.028 was used. The latter case ( M = 0.511) is
shown in Figure 6. The r 2 value was 0.98 in both cases. The
best fit of parameters M and M of Equation 10 was also tested
by nonlinear curve fitting resulting in M = 0.390 and
2
M = 0.520 (r = 0.99). Both 3/8 and the αM = 0.383 predicted
Figure 6. Observed and estimated (a) tree biomass (Equation 10) and
(b) tree height (Equation 9) as a function of basal diameter in Populus
deltoides trees grown in a dense stand. The WBE estimate for biomass
(a) was calculated with M = 0.38282 and M = 0.51136. The WBE estimate for height (b) was calculated with H = 2/3 and H = 3.188, and
statistical estimate with H = 0.9848 and H = 2.152.
TREE PHYSIOLOGY ONLINE at http://heronpublishing.com
8
NYGREN AND PALLARDY
by Equation 7 were within the confidence interval of the
best-fit M value (0.369–0.411), indicating that the scaling between stem basal radius and tree biomass predicted by the
WBE model held well for young P. deltoides trees.
Discussion
Because variation in vessel radius between successive links
was small, cross sections that were situated a few link orders
apart were analyzed. Although this improved the quality of
vessel analyses, it explains the poor fit of Equation 1 to the link
radius data. Each branching order in Populus deltoides contains links at different morphological levels and of different radii (Figures 1 and 2). Thus, the data including link radii at various non-contiguous levels was too noisy for curve fitting. It
has been observed that the cross-sectional area of the main link
formed in a branching point of P. deltoides accounts on average for 88% of the sum of cross-sectional areas of all links of
the branching point throughout a tree (Nygren et al. 2004).
The scaling parameter a for link radii could be estimated with
Equations 4 and 12, supporting the assumption that the failure
to fit Equation 1 was a result of the sampling method and not of
the architecture of P. deltoides. Further support was provided
by an unpublished dataset (Miaoer Lu, University of Missouri–Columbia), in which Equation 1 could be fit to data on
link radii before and after a branching point. Thus, use of the
value of a estimated from Equations 4 or 12 was considered
justified in the analysis of vascular allometry of P. deltoides.
The value of 0.223 for scaling factor b for vessel radii observed in P. deltoides was close to the predicted value of 1/6
that West et al. (1999a) claimed to minimize hydrodynamic resistance in the vascular system. The deviation in the value of b
in P. deltoides from the predicted value may have been caused
by the relatively small range of link orders observed in the
young trees sampled; the value of b predicted by Equation 5 is
0.192 for 29 branching generations, or the number of link orders in roots and shoots of open-grown trees and shoots of
stand-grown trees, and 0.201 for 21 generations in shoots of
open-grown trees. Equation 2 fit satisfactorily to whole-tree
data from lateral roots of Order –8, i.e., 8 branching generations down from the root collar, to highest leafy branches in
the open-grown tree dataset. This indicates that the whole water conducting network of the sampled trees possesses the
fractal properties assumed in the WBE model. This analysis
omitted fine roots or leaves that differ both functionally and
structurally from the water conducting network connecting
coarse roots to stem to branches. The original WBE model includes petioles in the vascular allometry, but not explicitly fine
roots (West et al. 1999a, Enquist 2002). However, including
the almost invariant petiole vessel radius at various link orders
where the leaves are situated in P. deltoides caused failure of
fitting of Equation 2 to field data.
The whole-tree fit of Equation 2 obscured the fact that vessel radius decreased with morphological level rather than with
link order (Figure 3, Table 1). One of the simplifying assumptions made in the WBE model is that “their [xylem vessels’]
diameters are constant within a branch segment but are al-
lowed to vary between segments, thereby allowing for possible tapering of xylem tubes from trunk to petiole” (Enquist
2002). Our data show that tapering occurs by morphological
level of plant organs and not between subsequent links as assumed in the WBE model constructions. This observation is
closer to the data reviewed by Zimmermann (1983), indicating
that hydraulic properties of vessels vary between organs,
rather than continuously within an organ. The WBE model includes an implicit assumption that all links are morphologically equivalent, i.e., of the same morphological level. In nature, this holds for Leeuwenberg’s model that has branches
only on top of a stem of variable length (Hallé et al. 1978).
Parameter b values approaching 1/6 have been observed in
some datasets on vessel tapering in plant shoots (Enquist
2003). In our study, tapering was observed in whole-tree analysis from lateral roots to leafy branches in the open-grown
trees but Equation 8 failed to fit the shoot data from standgrown trees (Figure 3). As far as we know, no other dataset
concerning the whole water conducting pathway from roots to
terminal branches exists. This conplicates the interpretation of
the apparent vessel tapering at the whole-tree level but not at
the shoot level in our study and at the shoot level in data reviewed by Enquist (2003). Thus, vessels appear to taper by a
factor of 1/6 at the whole-tree level, at least in some tree species, but the hydrodynamic significance of this factor remains
debatable as Mäkelä and Valentine (2006) have recently
shown that a b value approaching 1/6 does not result in invariant resistance to water flow in a vessel system. However, the
assumption of whole-tree vessel tapering by a factor of 1/6
may be an artifact that obscures step-wise decreases in vessel
radii by morphological level.
Equation 4 produced an apparently good fit to the data on
the relationship between link radius and water conductive area
(Figure 4) but underestimated the conductive area when link
radius was small. Systematic error within a part of a large
range of an independent variable is quite common in curve fitting (Mayer and Butler 1993) and can only be detected by examining model residuals. The systematic error was avoided
when water conductive area was related to link radius only
(Equation 12). Equation 12 describes a statistical relationship
that is not derived from the WBE model.
The better fit of the statistical Equation 12 than the WBE
model (Equation 4) may be due to the unrealistic assumption
of the WBE model that leaf petioles in all cases are situated at
one terminal branching level (West et al. 1997, 1999a). This is
an unreasonable assumption for P. deltoides, as well as for all
trees that do not follow the branching pattern of Leeuwenberg’s or Schoute’s model (Hallé et al. 1978). Thus, petiole
traits are unrelated to link traits at other branching levels,
which were up to 27 branching generations higher than the
lowest petiole order in our stand-grown trees and 19 generations higher in our open-grown trees.
Conductive area in a petiole varied little in either open- or
stand-grown trees. The relationship between number of leaves
above an analyzed cross section and conductive area in leafy
branches (Figure 5) indicates that leaf traits strongly affect the
architecture of terminal branches. This observation is also in
TREE PHYSIOLOGY VOLUME 28, 2008
VASCULAR ALLOMETRY IN A MONOPODIAL TREE
accordance with both the pipe model theory (Shinozaki et al.
1964) and the WBE model (Enquist 2002).
The scaling factor between link radii a estimated from
log–log transformation of Equation 12 was close to one in both
open- and stand-grown trees, implying that the area-preserving pipe model holds for link radii in P. deltoides (cf. Enquist
2002). The observation that the conductive area to total sapwood area ratio decreases from roots with largest vessel radii
to branches with smallest vessel radii (Table 2) is consistent
with the relationship between parameters a and b. The former
indicates that total link area is preserved at each branching
point, and the latter indicates that vessel area decreases
upward in a tree.
Both vessel diameter and the conductive area to sapwood
area ratio were significantly larger in woody parts of opengrown P. deltoides than stand-grown P. deltoides (Tables 1
and 2). Water supply has been observed to affect vessel diameter in Populus spp., with a tendency for wider vessels when
water supply is abundant (Schume et al. 2004). Soils of the
studied sites, which were situated close to each other, were
quite similar. Smaller vessel radii in the stand-grown trees may
indicate competition for water in the dense stand. The smaller
water conductive area in leaf petioles of stand-grown trees
may be associated with smaller mean area of leaf laminae in
the stand. Despite these observed differences, link radius and
water conductive area within a tree appeared to scale with
same parameters at both sites (Figure 4).
The relationship between stem basal radius and biomass
predicted by the WBE model (Equation 10) held well for
young P. deltoides growing in the dense stand (Figure 6).
Equation 10 fit the data well when M = 3/8. Because the value
of a estimated by Equation 12 is 1.028 ≈ 1, the exponent predicted by Equation 7 gives M = 3a/(2(a+3)) ≈ 3/8. Statistical
estimation of M by nonlinear curve fitting resulted in a value
that did not significantly differ from either estimate based on
the WBE model. Thus, the quarter-power scaling law seems to
fit well to the biomass–basal stem radius relationship in young
P. deltoides. The trees studied had no heartwood, although
self-pruning had already occurred in the dense stand. Heartwood formation may alter the observed relationships, but, if
observed in other fast-growing tree species, the quarter-power
law may have practical applications in the evaluation of shortrotation plantations.
The fit of the predicted relationship between stem basal diameter and tree height (Equation 6 resolved as Equation 9)
was not as good as that for biomass (Figure 6). The best-fit relationship was closer to a linear function than to a power function. Tree height is strongly affected by environment, and in a
dense stand, trees tend to grow tall quickly because of intense
competition for sunlight. The same holds for link length, making its estimation a difficult task in fractal analyses of tree
architecture (Salas et al. 2004).
It may be concluded that link radii in young P. deltoides
scale according to the area preserving pipe model. Vessel radii
appeared to taper by a power of about 1/6, as predicted by the
WBE model at the whole-tree level, but closer examination revealed that vessel radii varied between morphological levels
9
rather than continuously within the whole tree. Further, the apparent 1/6 power tapering was observed only in the opengrown tree data that also included the root vessels and covered
five morphological levels, whereas a tapering exponent could
not be determined from stand-grown tree data of three shoot
morphological levels. This suggests that the modeled 1/6
power tapering of vessels may be an artifact of applying Equation 2 to large datasets without considering morphological
variation between plant organs. Because it has recently been
shown that 1/6 power tapering does not imply invariant hydrodynamic resistance within the vascular system (Mäkelä and
Valentine 2006), it will be of interest to study the hydrodynamic or other biophysical significance of the variations in
vessel radii between plant organs. The variation with morphological level resulted in decreased vessel radius of P. deltoides
upward following water flow.
Our results are at odds with the basic WBE model assumption on the existence of a certain terminal level with constant
vessel radius. Although petiole vessel radius and total water
conductive area appeared to be rather invariant in P. deltoides,
leaves were not situated at one terminal level of branching
generations but at almost any shoot link order. This pattern
may have caused the systematic error in Equation 4 at small
link radii.
Vascular allometry as formalized in the WBE model appears to be useful for analyzing universal characteristics of
tree architecture. This is reflected in the fit of the quarterpower law for predicting tree biomass, which is of such exactitude that it is worth studying for practical forestry applications. However, the architectural diversity of trees likely
causes many deviations from universal scaling, which require
further study before the WBE model can be generally applied
to individual tree architecture and allometry.
Acknowledgments
We thank Victor Nieto for skillful vascular sampling and microscopic
measurements, Ryan Dowell for whole-tree data measurements, Prof.
Richard Guyette for access to the Dendrochronology Laboratory of
the University of Missouri– Columbia, and Prof. Annikki Mäkelä for
constructive criticism on an earlier manuscript version. Field work
was conducted while PN was working with the Department of Forestry of the University of Missouri– Columbia with funding from the
University of Missouri Life Sciences Mission Enhancement Program.
Funding was also provided through the University of Missouri Center
for Agroforestry under cooperative agreements AG-02100251 with
the USDA Agricultural Research Service Dale Bumpers Small Farms
Research Center, Booneville, AR and CR 826704-01-0 with the U.S.
Environmental Protection Agency. The results presented are the sole
responsibility of the co-authors and the University of Missouri and
may not represent the policies or positions of the Agricultural Research Service or the Environmental Protection Agency.
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