Tree Physiology 18, 259--264
© 1998 Heron Publishing----Victoria, Canada
Differences in root longevity of some tree species
K. E. BLACK,1 C. G. HARBRON,2 M. FRANKLIN,2 D. ATKINSON1 and J. E. HOOKER3
1
The Scottish Agricultural College, West Mains Road, Edinburgh EH9 3JG, U.K.
2
Biomathematics and Statistics Scotland (BioSS), Rowett Research Institute, Greenburn Road, Bucksburn, Aberdeen AB21 9SB, U.K.
3
Soil Biology Unit, Land Resources Department, SAC, Doig Scott Building, Craibstone Estate, Aberdeen AB21 9TQ, U.K.
Received December 7, 1995
Summary Although the importance of root production and
mortality to nutrient fluxes in ecosystems is widely recognized,
the difficulties associated with root measurements have limited
the availability of reliable data. We have used minirhizotrons
and image analysis to measure root longevity of Prunus
avium L., Picea sitchensis (Bong.) Carrière, Acer pseudoplatanus L. and Populus × canadensis cv. Beaupre directly in
cohorts of roots. Major differences in the longevity of roots
among species were identified. For example, 40% of Prunus
avium roots but only 6% of Picea sitchensis roots survived for
more than 14 days. Survival analysis of cohorts of roots of
Prunus avium and Populus × canadensis revealed differences
in the distribution of longevity among cohorts. Genetic, biotic
and abiotic factors that may influence longevity are discussed.
Keywords: Acer pseudoplatanus, minirhizotrons, nutrient cycling, Picea sitchensis, Populus × canadensis cv. Beaupre,
Prunus avium, survival analysis.
Introduction
Both the production and longevity of fine roots are major
factors affecting carbon and nutrient fluxes within terrestrial
ecosystems. This is because roots are not only a sink for carbon
and nutrients, but are also a source of nutrients for the plant.
Although the importance of roots in nutrient cycling has been
recognized for many years (e.g., Joslin and Henderson 1987),
technical limitations have prevented accurate quantification of
the importance of root mortality in nutrient cycling. The absence of these data has resulted in the development of nutrient
cycling models (reviewed by Schimel et al. 1990) that do not
take full account of the role of roots. In the past, the soil has
presented the major obstacle in obtaining accurate measurements of roots because it limits accessibility. Traditional sampling methods, e.g., repeated soil coring, have been frequently
employed (reviewed by Böhm 1979) and have produced data
suggesting that roots can account for a significant input of
carbon and nitrogen to soils. For example, Vogt et al. (1986)
estimated that root mortality adds 18 to 58% more nitrogen to
soils than litter fall. However, there is a major limitation to
these methods that prevents reliable quantification. Because
root production and mortality can occur simultaneously within
a sampling interval, mortality rates are greatly underestimated
(Kurz and Kimmins 1987, Hendrick and Pregitzer 1992a).
Furthermore, in most ecosystems, the soil coring approach is
subject to high sampling variability (Singh et al. 1984) and this
drawback is especially evident in trees because of the asymmetric nonrandom distribution of tree root systems (Hendrick
and Pregitzer 1992b).
The only reliable method for quantifying root dynamics and
longevity is to observe roots nondestructively in situ over time.
This is now possible as a result of recent developments in video
technology. Minirhizotrons (transparent plastic or glass tubes)
are inserted in the soil and roots viewed at the soil--tube
interface with a miniature television camera. The usefulness of
this technique for measuring root longevity has recently been
demonstrated by Hendrick and Pregitzer (1993) for sugar maple and Hooker et al. (1995) for poplar.
The aim of the present study was to measure root longevity
for a range of different tree species by using a similar approach.
The data obtained were subjected to survival analysis (Lee
1992). Survival analysis is a powerful technique for studying
data that are censored. Furthermore, it is more appropriate than
the simpler techniques employed in other root longevity studies
(e.g., Hendrick and Pregitzer 1993, Reid et al. 1993, Hooker
et al. 1995), because it makes more complete use of the data and
permits both comparisons of root longevity between tree species and a description of temporal change within species.
Materials and methods
Experimentation
One-year-old seedlings of Prunus avium L. (cherry), Populus
× canadensis cv. Beaupre (poplar), Picea sitchensis (Bong.)
Carrière (Sitka spruce) and Acer pseudoplatanus L. (sycamore) were grown in nursery compost in 22.9-cm (6.2-liter)
pots. There were five replicates per species. Plants were maintained in an unheated greenhouse in natural light and watered as
necessary. A 250-mm long minirhizotron tube (a clear Perspex
(Plexiglas) tube with an internal diameter of 50 mm and an
external diameter of 60 mm) marked with guidelines was inserted vertically in each pot. The portion of the tube extending
above the surface of the soil was covered to prevent light
penetration. The inner wall of the minirhizotron tube was
imaged every 7 days with a miniaturized color television
260
BLACK ET AL.
camera (Bartz Technology, Santa Barbara, CA). Images were
stored on videotape. To restore the camera each time to the
same position in the soil, visual reference points were marked
on the inner wall of the tube, enabling the observation of
individual roots over time. Binary overlays of the roots in each
frame were generated and labeled with an appropriate identifier, using the image analysis software ROOTS (Hendrick and
Pregitzer 1992b, Hooker and Atkinson 1992).
The longevity of roots was studied by the cohort analysis
method (Hendrick and Pregitzer 1993). A cohort consists of
those roots produced within a defined period, here usually
7 days. The progress of the roots in each cohort was followed
for a defined period, here for 9 weeks. For example, Cohort
One consists of the roots produced in the week between June
27 and July 4, and the progress of these roots was followed
until September 5. For each species, the data presented were
derived from 10 successive cohorts of roots produced in successive weeks between June 27 and September 5. Each cohort was
followed for 9 weeks.
Statistical analysis
Comparisons were made both among species and among cohorts of the same species of the proportion of roots surviving
until the end of the study, i.e., for at least 63 days, by fitting
logistic models (McCullagh and Nelder 1989), which allowed
for both pot effects and species or cohort effects. Allowance
was also made for a dispersion factor greater than unity.
Survival analysis
 t

S(t) = exp −∫ h(u )du .
 0

(3)
Thus, the probability of an individual root surviving beyond
time t is related to the accumulated hazard up to time t.
Because we had no prior knowledge of the distribution of
root longevities, we required a flexible distribution that permits modeling of a wide range of survival distributions. Various distributions fit the basic requirement that they cover the
continuous scale for time. These include the Exponential,
Weibull, Gamma and Lognormal distributions (see Appendix).
The Exponential distribution is defined by a single parameter,
whereas the others are each described by two parameters
allowing them greater flexibility. It is useful to note that both
the Weibull and Gamma distributions include the Exponential
distribution as a special case. For the purposes of this study, we
used the Weibull distribution. The Weibull distribution is a
survival distribution described by two parameters, a scale
parameter θ and a shape parameter γ. Its survival function can
be expressed as:
γ
 t 
S (t) = −    ,
 θ  
(4)
and its hazard function as:
The longevities of some roots were unknown; we only have the
information that they survived for at least the duration of the
study. Such data are said to be censored. Survival analysis is a
powerful technique for studying censored data (Lee 1992). In
this study, censoring took a simple form, namely that no
observations were recorded after 63 days. However, survival
analysis can also cover more complex patterns of censoring;
for example, it can allow for the accidental destruction of some
roots.
A survival distribution is essentially a probability distribution of the lifetimes of the objects under examination. In our
study, the distribution is of longevity of a population of roots.
Two informative functions to observe when examining the
properties of a particular survival distribution are the survivor
function S(t), which defines the probability of an individual
surviving until at least time t, and the hazard rate function h(t),
which defines the instantaneous probability of death for an
individual alive at time t. A survival distribution with probability density function for root lifetimes f(t) and cumulative
probability density function F(t) can be expressed as:
S (t) = P (T ≥ t) = 1 − F (t),
h (t) = P(t + dt ≥ T > t | T > t) /dt =
These two functions are related in the following manner:
(1)
f(t)
,
S (t)
(2)
where P is probability, T is a random variable describing a
root’s longevity and dt is a sufficiently small time interval.
γ t 
h (t) =  
θθ
γ−1
.
(5)
The scale parameter θ is the main determinant of the degree
of hazard and thus the average life span. Large values of θ
correspond to low hazards, which is equivalent to a large
proportion of individual roots surviving a long time, whereas
small values of θ correspond to high hazards and a rapidly
decaying survivor function with very few roots living for a
long time. When γ = 1, which corresponds to the exponential
distribution, θ is the mean survival time.
The shape parameter γ determines the change in the degree
of hazard over time. When γ = 1, the hazard is constant, i.e.,
the probability of an individual that is alive at the start of a
fixed interval failing to survive until the end of that interval
does not change. A value of γ greater than 1 corresponds to an
increasing hazard rate, where older individuals are more likely
to die within a given time period than younger individuals. A
value of γ less than 1 corresponds to a decreasing hazard rate,
for example, in situations where young individuals are vulnerable, but this risk declines with age. Increasing risk with time
tends to result in survival time distributions with a low coefficient of variation (cv), because with low risks few roots die
young and with high risks few roots survive to extreme old age.
Conversely, decreasing risks tend to give distributions with a
high cv. As is shown in the Appendix, there is a direct relationship between the cv and one of the parameters in the two-pa-
TREE PHYSIOLOGY VOLUME 18, 1998
DIFFERENCES IN TREE ROOT LONGEVITY
rameter distributions. For the Weibull distribution, there is a
direct (inverse) relationship between the cv and γ.
For each tree species, data for cohorts were grouped as
necessary, and distributions were fitted to the data. The parameters of the distributions were estimated by maximum
likelihood (e.g., Lee 1992) using the Genstat 5 statistical package (The Numerical Algorithms Group Ltd., Oxford, U.K.).
Estimates obtained from each of these fitted distributions were
treated as coordinates in a graph where estimates for the mean
formed the x-axis and estimates for the coefficient of variation
formed the y-axis. Successive cohorts are linked to show how
the distribution of root longevities changed over time. Cohorts
with similar distributions of root longevity appear close together, whereas those with different distributions are separated. The position where a cohort is plotted defines the two
most important features of the pattern of root longevities
within that cohort. The plot thus allows comparison between
different species and also allows temporal changes in longevity
within a species to be observed. Contours of equal values for
θ have been included in the plot to allow changes in the
parameter value to be monitored. Also, as shown in the Appendix, there is a direct relationship between the value of the
coefficient of variation (cv) and γ, so γ is used to provide an
alternative scale for the y-axis.
Similar graphs can be drawn for each of the two-parameter
distributions. However, in the presence of censoring, the various distributions will yield differing estimates of the mean and
coefficient of variation because of the assumptions used about
the survival times to be allocated to the censored observations.
Results
Figure 1 shows cumulative mortality curves for each species
with all cohorts combined. On the logarithmic scale used,
periods of constant risk are indicated by straight lines. A total
of 235 cherry roots were monitored with 18 surviving for
63 days or more. The corresponding values for the other three
species were poplar 219 and 80, Sitka spruce 32 and 17, and
sycamore 29 and 7. Among the species, cherry roots were the
most short-lived with over 40% surviving for less than 14 days
and fewer than 8% surviving for longer than 63 days. Cherry
had significantly fewer roots surviving in excess of 63 days
than any of the other species (P < 0.05). In contrast, Sitka
spruce roots survived for much longer periods, with over 53%
living longer than 63 days and only 6% failing to survive for at
least 14 days. All of the species differed significantly in the
proportion of roots surviving for at least 63 days.
There were insufficient Sitka spruce or sycamore roots to
detect differences among cohorts in the proportion of roots
surviving to 63 days. In cherry, no significant differences were
detected among cohorts in the proportion of roots surviving to
63 days (Table 1). (A more sensitive test for cherry survival
would be to use a shorter survival period.) In poplar, there were
marked differences (P < 0.01) among cohorts----survival being
higher for later cohorts than for earlier cohorts. This shows that
temporal changes occurred in root longevity during the study,
261
Figure 1. Root survival curves for each species with all cohorts
combined.
which extended from the middle of the growing season until
dormancy.
Survival analysis provided further information about the
temporal changes. Figure 2 shows the mean cv plot for cherry
and poplar roots. The estimate for θ is given by the position of
the plotted point with respect to the dashed contour lines.
Figure 2 shows that, in the earlier cohorts, the mean longevities
of cherry and poplar roots were similar. However, for later
cohorts, the longevity decreased for cherry but increased for
poplar.
Estimates for the parameters θ and γ, together with their
standard errors, are given for each of the cohort sets in Table 1.
For Sitka spruce and sycamore, the data from all cohorts have
been pooled. For cherry and poplar, estimates are given also
for the cohort groups 1--4, 5--6, 7--8 and 9--10. For the overall
pooled data (Cohorts 1--10), the two-parameter distributions
showed no improvement over the simple exponential for
cherry and sycamore. Therefore, for these two species it is
reasonable to accept that the risks were constant over the
survival period. For poplar, the risks were initially as high as
for sycamore but then declined (see Figure 1). As a consequence, the ‘least good’ fit for poplar was provided by the
exponential distribution (but even here the deviations from the
distribution are not statistically significant, P > 0.05). There
was a small but not statistically significant (P > 0.05) improvement from fitting the Gamma or Weibull distribution as opposed to the Exponential distribution; however, the best fit was
given by the Lognormal distribution. For Sitka spruce, like
poplar, the risks appear to decrease for older roots with other
distributions giving slight but not statistically significant
(P > 0.05) improvements in fits from the two-parameter distributions.
Discussion
In young trees, root life can be short. For example, in cherry,
fewer than 60% of roots survived more than 14 days (Figure 1).
TREE PHYSIOLOGY ON-LINE at http://www.heronpublishing.com
262
BLACK ET AL.
Table 1. Estimated parameters θ and γ for the Weibull distribution for survival times (days) of roots from four tree species: cherry, poplar, Sitka
spruce and sycamore, with the standard errors of the estimates plus the estimated mean and coefficient of variation (cv). Estimates are given for
each species totalled over all cohorts (1--10) and, in cherry and poplar, for sets of cohorts. Also given is the percentage of roots surviving ≥
63 days.
Cohort
θ (SE)
γ (SE)
Mean
cv
Proportion (%)
surviving ≥ 63 days
Cherry
C1--4
C5--6
C7--8
C9--10
C1--10
36.8 (8.2)
32.2 (2.6)
24.9 (3.1)
15.8 (2.6)
26.4 (1.7)
1.04 (0.24)
1.53 (0.16)
0.89 (0.09)
1.05 (0.16)
1.05 (0.06)
36.2
29.0
26.4
15.5
25.9
0.96
0.67
1.13
0.95
0.95
11.7 (6.4)
5.3 (2.5)
11.1 (3.2)
2.6 (2.5)
7.7 (1.7)
Poplar
C1--4
C5--6
C7--8
C9--10
C1--10
35.2 (4.5)
21.6 (5.2)
47.8 (15.1)
102.0 (15.6)
61.7 (6.1)
1.25 (0.16)
0.68 (0.12)
0.83 (0.21)
1.26 (0.91)
0.88 (0.7)
32.8
28.1
52.8
94.8
65.7
0.80
1.51
1.21
0.80
1.14
18.1 (5.4)
16.2 (5.9)
31.1 (9.1)
53.7 (5.4)
35.8 (2.6)
Sitka spruce
C1--10
85.5 (18.3)
1.38 (0.33)
78.1
0.73
53.1 (10.3)
Sycamore
C1--10
45.0 (8.1)
1.19 (0.24)
42.4
0.84
24.1 (8.0)
Figure 2. Mean cv plot for
cherry and poplar roots showing
changes with cohort group as detailed in Table 1.
Furthermore, major differences exist in the longevity of roots
of different species as indicated by the finding that 94% of
Sitka spruce roots survived for more than 14 days. Survival
analysis provided detailed information on temporal changes in
the distribution of root longevities (Figure 2). Temporal
changes were more complicated than a simple increase or
decrease in average root longevity and occurred often, whereas
the mean lifetimes remained approximately constant. These
data provide the first evidence both for genetically determined
differences in root longevity among species and also for significant changes in longevity over time.
Survival analysis makes a more complete use of the data
within each cohort by using the age at death of all the roots
rather than a simple indicator of ‘‘survived longer than’’ where
the arbitrary choice of the cut-off time may strongly affect the
results. The summary obtained for each cohort, i.e., the shape
and scale parameters, thus provides a more complete description of the pattern of root longevities within the cohort. From
TREE PHYSIOLOGY VOLUME 18, 1998
DIFFERENCES IN TREE ROOT LONGEVITY
the fitted parameters, θ and γ, estimates of other statistics
describing the cohort can also be made. For example, the mean
longevities and cv, used as axes in Figure 2, could be replaced
by the median and inter-quartile ranges, respectively. The
technique thus provides a comprehensive analysis of root data
of this type.
Other reports of tree root longevity measured by direct
methods are rare. However, our values for poplar are similar to
those obtained by Hooker et al. 1995 for the same species
(using the same method, although growth conditions in the two
studies were quite different). Hendrick and Pregitzer (1992a)
also used a similar technique to measure root longevity in a
sugar maple forest. They assumed most roots to be of Acer
saccharum Marsh. and concluded that mean survival ranged
from 5.5 to 10 months. Fahey and Hughes (1994) used in situ
screens to estimate longevity of roots produced in early summer (mid-June to early July) in a northern hardwood forest and
obtained average values of 8 to 10 months. Thus, both of these
studies report root longevities significantly greater than the
values we obtained. However, in a study using a root observation laboratory, Reid et al. (1993) determined that 40% of kiwi
fruit roots survived for less than 28 days, a value that is
comparable to the values we obtained with different species.
The evidence indicates that large differences in root longevity exist among species. However, the differences we observed
among species were not as large as the differences between our
data and those of Hendrick and Pregitzer (1992a) and Fahey
and Hughes (1994), which indicate lifetimes 10 to 20 times
greater than we observed. The reasons for these large differences are likely to be complex and may include genetically
determined factors, prevailing environmental conditions and
the physiological status of the trees. Moreover, the temporal
changes in longevity that we observed indicate that it is possible to measure large differences in root longevity at different
points in time. There have also been suggestions that temperature may influence the longevity of roots (Hendrick and
Pregitzer 1993).
An alternative explanation is that root longevity is more
closely linked to the physiological status of the rest of the tree
than to direct effects of the environment on belowground
processes and that, although there may be a link between
temperature, day length and perhaps other environmental variables on belowground processes, root responses are indirectly
mediated through the influences of aboveground processes.
There is no information on whether root longevity changes
with the age of the tree or its relationship with the age of the
root. Moreover, recent data by Hooker et al. 1995 have shown
that colonization by arbuscular mycorrhizal fungi can reduce
root system longevity. Thus, given that temporal changes in
colonization occur (McGonigle and Fitter 1990, Sanders and
Fitter 1992), arbuscular mycorrhizal fungi may, at least in part,
drive the temporal changes in root longevity.
We conclude that, in young trees, roots can be relatively
short-lived, longevity can change over time, and there are
differences in longevity among species. Moreover, the short
life spans observed suggest that the importance of root turnover has often been understated and can represent a major
263
nutrient and carbon flux. This conclusion should contribute to
better understanding nutrient and carbon cycling within treebased ecosystems.
Acknowledgments
Both SAC and BioSS receive financial support from The Scottish
Office Agriculture Environment and Fisheries Department.
References
Böhm, W. 1979. Methods of studying root systems. Springer-Verlag,
Berlin, 188 p.
Fahey, T.J. and J.W. Hughes. 1994. Fine root dynamics in a northern
hardwood ecosystem, Hubbard Brook Experimental Forest, NH.
J. Ecol. 82:533--548.
Hendrick, R.L. and K.S. Pregitzer. 1992a. The demography of fine
roots in a northern hardwood forest. Ecology 73:1094--1104.
Hendrick, R.L. and K.S. Pregitzer. 1992b. Spatial variation in tree root
distribution and growth associated with minirhizotrons. Plant Soil
143:283--288.
Hendrick, R.L. and K.S. Pregitzer. 1993. Patterns of fine root mortality
in two sugar maple forests. Nature 361:59--61.
Hooker, J.E. and D. Atkinson. 1992. Application of computer-aided
image analysis to studies of arbuscular endomycorrhizal fungi effects on plant root system morphology and dynamics. Agronomie
12:821--824.
Hooker, J.E., K.E. Black, R.L. Perry and D. Atkinson. 1995. Arbuscular mycorrhizal fungi induced alteration to root longevity of poplar.
Plant Soil 172:327--329.
Joslin, J.D. and G.S. Henderson. 1987. Organic matter and nutrients
associated with fine root turnover in a white oak stand. For. Sci.
33:330--346.
Kurz, W.A. and J.P. Kimmins. 1987. Analysis of some sources of error
in methods used to determine fine root production forest ecosystems: a simulation approach. Can. J. For. Res. 17:909--912.
Lee, E.T. 1992. Statistical methods for survival data analysis. John
Wiley and Sons, New York, 482 p.
McCullagh, P. and J.A. Nelder. 1989. Generalised linear models.
Chapman and Hall, London, 511 p.
McGonigle, T.P. and A.H. Fitter. 1990. Ecological specificity of vesicular-arbuscular mycorrhizal associations. Mycol. Res. 94:120-122.
Reid, J.B., I. Sorensen and R.A. Petrie. 1993. Root demography in
kiwifruit (Actinidia deliciosa). Plant Cell Environ. 16:949--957.
Sanders, I.R. and A.H. Fitter. 1992. The ecology and functioning of
vesicular--arbuscular mycorrhizas in co-existing grassland species.
I. Seasonal patterns of mycorrhizal occurrence and morphology.
New Phytol. 120:517--524.
Schimel, N.C., W.J. Parton, T.G.F. Kittel, D.S. Ojima and C.V. Cole.
1990. Grassland biogeochemistry: links to atmosphere processes.
Clim. Change 17:13--25.
Singh, J.S., W.K. Lauenroth, H.W. Hunt and D.M. Swift. 1984. Bias
and random errors in estimators of net root production: a simulation
approach. Ecology 65:1760--1764.
Vogt, K.A., C.C. Grier and D.J. Vogt. 1986. Production, turnover and
nutritional dynamics of above- and belowground detritus of world
forests. Adv. Ecol. Res. 15:303--307.
Appendix
We briefly report some properties of four distributions that
could prove useful in analyzing survival data. For the simplest
TREE PHYSIOLOGY ON-LINE at http://www.heronpublishing.com
264
BLACK ET AL.
distribution, the Exponential, the risk (or hazard) is constant at
all times. The Weibull and the Gamma distributions allow the
risk to increase or decrease monotonically, the latter having a
nonzero asymptotic limit. Both can be seen as generalizations
of the exponential. The Lognormal has differing properties and
its usefulness is based on the logarithms of survival times
being normally distributed.
(b) The cv is not affected by θ. It decreases as γ increases.
(c) When γ = 1, the Weibull distribution is equivalent to the
Exponential distribution.
Gamma distribution
f(t) =
Exponential distribution
f(t) =
1
 t
exp −  ,
θ
 θ
h (t) =
has
1 t
Γ(γ)  θ 
1 t
θΓ (γ)  θ 
γ
∑
h (t) =
1
1
,
θ
t
θ 
 
 −t 
1
exp   ,
θ
θ
γ−1
r−1
,
/ Γ(r)
mean = θγ , SD = θγ1/2 , cv = γ− 1/2.
mean = θ, SD = θ, cv = 1.
Weibull distribution
γ t 
γ−1
γ−1
f(t) =  
θθ
γ t 
h (t) =  
θ θ 
γ
γ
γ−1
 t 
 t 
t 1
exp −   = γ  
exp −   ,
 θ 
 θ  
θ  θ
γ−1
(a) The hazard function increases from 0 to 1/θ as t increases
from 0 if γ > 1. It decreases from infinity to 1/θ if γ < 1.
(b) The cv is not affected by θ. It decreases as γ increases.
(c) When γ = 1, the Gamma distribution is equivalent to the
exponential distribution.
Lognormal distribution
When transformed to natural logarithms the distribution has
mean = log e(θ).
The hazard function, which involves the normal integral is
not given.
,
mean = θΓ(γ−1 + 1 ),
SD = θ(Γ(2γ + 1) − (Γ (γ + 1)) ) ,
γ 2 
γ 2
mean = θ exp   , SD = θ exp  (exp (γ 2) − 1)1/2 ,
2
2
cv = (Γ(2γ−1 + 1) − (Γ(γ−1 + 1 ))2)1/2 /Γ(γ−1 + 1),
cv = (exp (γ 2) − 1)1/2.
−1
−1
2 1⁄ 2
where Γ(γ− 1 + 1) denotes the gamma function.
(a) The hazard function increases from 0 to infinity if γ > 1. It
decreases from infinity to 0 if γ < 1.
(a) The hazard function increases from θ to infinity as t increases from zero.
(b) The cv is not affected by θ. It increases as γ increases.
TREE PHYSIOLOGY VOLUME 18, 1998