Annals of Botany 83 : 335–345, 1999
Article No. anbo.1999.0829, available online at http:\\www.idealibrary.com on
Analysis of Distribution of Root Length Density of Apple Trees on Different
Dwarfing Rootstocks
H. N. D E S I L V A*, A. J. H A L L*, D. S. T U S T I N† and P. W. G A N D AR*‡
* The Horticulture and Food Research Institute of New Zealand, PriŠate Bag 11030, Palmerston North,
New Zealand and † PriŠate Bag 1401, HaŠelock North, New Zealand
Received : 20 July 1998
Returned for revision : 8 October 1998
Accepted : 8 December 1998
This paper considers statistical analyses for comparing the distribution of root length density (RLD) of apple trees
under different rootstocks and tree spacing. The source data included RLD values (cm cm−$) measured by soil coring
the root systems of eight trees in each of two seasons. We formulated a regression model which assumed the RLD
dropped exponentially with soil depth, and this relationship varied with the radial distance from the tree. The model
fitted to the log transformed mean data described the RLD distribution well. Young trees (5-year-old) of M.26 (semidwarf) and MM.106 (semi-vigourous) had a higher mean RLD and showed a more layered vertical distribution,
compared with trees of the dwarf Mark rootstock. Differences among rootstocks were not evident in older (9-yearold) trees. In general, young root systems were more bowl shaped, whereas older trees had a higher RLD further away
from the tree trunk. RLD is a positive and continuous variable except for the possibility of an excess of exact zeros.
A generalized linear model with a Poisson-gamma type distribution allows modelling of individual RLD data with
zeros contributing to parameter estimation. It does not, however, provide simplicity of biological interpretation. In
this paper we present a model that assumes the realization of RLD data is due to a Bernoulli and an exponential
process. The fitting of the Bernoulli-exponential model by maximum likelihood is illustrated, and further
generalization suggested.
# 1999 Annals of Botany Company
Key words : Malus domestica (Borkh.), Fuji, rootstock, root system, soil core sampling, Bernoulli–exponential model.
INTRODUCTION
Fine roots are the main components of the root system
through which plants absorb water and nutrients. These
relatively thin roots, with a high specific root length (length :
dry weight ratio), form the younger parts of the root system.
Fine roots are unsuberized and have a high permeability
compared to older ones. In apple trees, these roots are
generally
1n0 mm in diameter. Because of its functional
importance many researchers have investigated the distribution of fine root length density (RLD ; cm cm−$) within
the root zone. Atkinson (1980) has provided a review of
many studies in this area, involving different tree species.
The presence of fine roots in the soil volume occupied by
the root system will obviously depend on the geometry of
the structural root system. Tree spacing and type of
rootstock are two important orchard management practices
known to influence the density of fine roots in tree species
(Atkinson, Naylor and Coldrick, 1976 ; Atkinson, 1980). It
is plausible that these effects result from changes in the
geometry of the structural root system, and\or differential
production rates of fine roots. More recently Fernandez,
Perry and Ferree (1991, 1995), using the trench profile
method, found that nine similarly-aged apple rootstock
clones grown on a Malette sandy loam at Michigan, could
be classified into three groups on the basis of numbers of
‡ Current address : Ministry of Research Science and Technology,
PO Box 5336, Wellington.
0305-7364\99\040335j11 $30.00\0
total roots observed and their size categories. However, in
Ohio on a Canfield silt loam with a fragipan, only one of the
same nine rootstock clones was distinguishable from the
others. This implies that the rootstock effect is dependent on
soil type. The RLD distribution also varies depending on
age. Soil coring studies with kiwifruit (Gandar and Hughes,
1988) and non-dwarf apple trees (Hughes and Gandar,
1993) have shown younger root systems to have a semielliptic bowl-shaped structure. In contrast the older root
systems exhibited a more layered structure. Soil moisture
content and irrigation can also affect root distribution
(Levin, Assaf and Bravado, 1979 ; Neilson et al., 1997),
although these factors are not considered in this study.
The most commonly used method for estimating fine root
length density involves soil coring. One disadvantage of
coring is that because of high variability in the data
substantial sampling is required to estimate differences
between root systems with reasonable accuracy. This may
be particularly true for young apple trees which have sparse
root systems (Atkinson, 1980). Gandar and Hughes (1988)
introduced the term ‘ occupancy ratio ’ to describe soilcoring data. This was defined as the proportion of the soil
volume with RLD greater than a given value. Absolute
values of ‘ occupancy ratio ’ are likely to be very small and
are of little interest. Relative measures based on core
samples of fixed dimensions, however, can be useful for
comparing different root systems. One drawback of the
‘ occupancy ratio ’ type description is that in any analysis
that follows, the information on the positive RLD values
# 1999 Annals of Botany Company
336
De SilŠa et al.—Analysing Root Length Density of Apple Trees
and the zeros are handled quite separately. In terms of
problems associated with statistical analysis, soil coring
data have properties similar to physical phenomena such as
rainfall, where the measurement is positive and continuous
except for the possibility of exact zeros when it does not
occur. Such data cannot usually be transformed to normality
by power transformation, and even so the information in
zeros is not incorporated fully in the analysis. Smyth (1996)
approached the problem through regression modelling using
the established framework of generalized linear models, but
with an exponential family distribution between the Poisson
and gamma (Tweedie, 1984 ; Jorgensen, 1987). Although a
Poisson-gamma generalized linear model provides some
statistical rigour, it does not provide the same simplicity in
the biological interpretation as evidenced in the ‘ occupancy
ratio ’ type description of data.
The aims of this paper are : (1) to estimate tree-spacing
and rootstock effects on the distribution of RLD of apple
trees, by analysing two seasons’ root coring data using
standard methods ; and (2) to develop and illustrate the
fitting of a model for the analysis of coring data based on
assumptions of theoretical density functions for the underlying processes that may have generated the data.
STUDY DESCRIPTION
Experimental material
The root distribution study was first carried out in late
November 1993 on a research orchard block in Hawkes
Bay, New Zealand (39m40h S, 176m53h E), on 5-year-old
apple trees of cultivar ‘ Fuji ’. The trees were part of a larger
spacingirootstock study, and were chosen because they
represented a range of rootstock vigour and tree size. The
trees had been grafted onto either MM.106 (semi-vigorous),
M.26 (semi-dwarf), or Mark (dwarf) rootstocks and planted
at two different between-rowiwithin-row spacings (5i3 m
or 4i2 m). The planting was arranged in a randomized
complete block design, with two replicate plots. Each plot
consisted of 15 trees planted in a three rowifive tree array,
so that the middle three trees of the middle row were
completely guarded. The soil type of the site was a Twyford
Sandy Loam, a typical horticultural soil in the Hastings
region. The soil has no impervious layering effects, but is
routinely drained to a depth of 1n2 m using tile drainage.
The experimental plot area was set up for supplementary
irrigation using mini-sprinklers. The sprinkler pattern
ensured the volume of water delivered was uniform across
all plots irrespective of planting density. A 0n75–1 m wide
herbicide strip was maintained on either side of each tree
row. Trees on all rootstocks were managed similarly using
Slender Pyramid tree management practices (Tustin et al.,
1990). Pomological performance of rootstockispacing
combinations has been reported elsewhere (Tustin et al.,
1993). In general, cropping was abundant on all treatments.
Any indirect effects on root distribution due to any
differences in the cropping history are likely to be
insignificant compared with the rootstock and spacing
effects. Two replicate trees, one from each plot, were
selected for root system sampling from each of the following
four treatment groups : Mark, 5i3 (tree 1–2) ; Mark, 4i2
(tree 3–4) ; MM.106, 5i3 (tree 5–6) ; M.26, 4i2 (tree 7–8).
Root sampling was repeated in early December 1997 on the
same block, but with different trees belonging to the same
treatment groups. Sampling was carried out in late spring in
both years, hence root system conditions were generally
similar on an annual growth cycle basis. All samples were
collected from guarded trees. Summary statistics of trees
belonging to the four treatment groups are presented in
Table 1.
Data collection
Soil cores were taken from the rooting volumes of each of
the eight trees selected in a given season. Here, the available
rooting volume was defined as the area occupied by a tree to
a depth of 1 m of soil. In the 1993 sampling, eight cores of
diameter 46 mm to 1 m depth were taken from several
points within the 4 mi2 m available area, for trees planted
at the close spacing. A further eight cores were taken from
the remaining 0n5 m wide outer rectangular area for the
wider 5i3 m spacing. The cores were positioned to
representatively sample the whole occupied area of the tree,
but were otherwise randomly placed. In 1997, the rectangular area around the tree was stratified into 1i1 m
quadrants and cores were positioned about the middle of
each square to provide a sample of cores (n l 8) with a
more regular spread. For the wider spacing, eight additional
cores were sampled as before so that sampling densities
were similar for the two spacings (eight cores for 8 m# Šs. 16
cores for 15 m#).
An engine-powered concrete breaker was used to drive
coring tubes into the soil and a tripod and winch were used
to extract them as described by Welbank and Williams
(1968). In 1993, five or six random 100 mm length core
samples were chosen from each core to represent the entire
1 m of core length. The position of each core was specified
by the angle (clockwise) from the north facing row direction
(θ), and the horizontal radial distance from the trunk to the
sample point (r). Individual core samples within a given core
were specified by the depth (z) to mid-point. In the 1997
study, 50 or 100 mm was discarded from top of each core
and they were then cut into 100 mm lengths ; five alternative
lengths starting from the top were taken. The sampling plan
ensured that each tree had equal numbers of cores
commencing at the two different depths. The result was a
regular sampling of soil depth, with sampling mid-points at
either 0n1, 0n3, 0n5, 0n7, 0n9 m or 0n15, 0n35, 0n55, 0n75
and 0n95 m.
Roots were extracted using a root washing machine
designed by Smucker, McBurney and Srivastava (1982).
Roots from each core sample were then separated into two
fractions : (1) roots less than 1 mm diameter (‘ fine ’) ; and (2)
roots greater than 1 mm diameter (‘ woody ’). The total
length of fine roots in each core sample was measured using
an automatic root-length scanner. The woody fraction was
oven dried and its mass recorded. Root length density
(RLD) was calculated by dividing the total root length of
fine roots by the volume of the core sample (166 cm$). Only
the RLD data are presented in this paper.
De SilŠa et al.—Analysing Root Length Density of Apple Trees
337
T     1. AŠerage Šalues of ‘ Fuji ’ apple trees sampled for root distribution by soil coring, during spring of 1993 and 1997
1993
Rootstock*
Mark
Mark
MM.106
M.26
1997
Tree
spacing
(m)
Height
(m)
Mean
spread (m)
TCA†
(cm#)
Crown
vol. (m$)
Height
(m)
Mean
spread (m)
TCA
(cm#)
Crown
vol. (m$)
5i3
4i2
5i3
4i2
2n6
2n5
5n8
4n8
2n4
2n1
3n4
2n7
27n7
25n2
104n9
65n9
3n3
2n2
19n9
10n5
3n2
2n8
5n6
4n3
2n5
2n2
3n4
2n7
76n6
43n9
220n5
140n1
7n8
5n0
24n5
11n2
*Mark is a dwarf rootstock, and M.26 and MM.106 are semi-dwarf and semi-vigourous, respectively.
†Trunk cross sectional area.
Trees were 5 years old in 1993.
EXPLORATORY ANALYSES
The root length density (RLD) at a point corresponding to
the centroid of a core sample was defined as the total length
of fine roots (cm) per unit volume of soil (cm$) sampled.
RLD values of core samples taken in the 1993 season were
relatively low, ranging up to only 0n60 cm cm−$, and with
75 % of values lying below 0n12. Apple trees were only 5
years old then, and the root mass was unlikely to be fully
developed. In contrast, trees sampled 4 years later had RLD
values ranging up to 5n1 cm cm−$, with an upper quartile of
0n9 ; almost a ten-fold increase in density.
Previous studies have shown no effect of angular direction
on root distribution (Gandar and Hughes, 1988 ; Hughes
and Gandar, 1993). Assuming the root system has a radial
symmetry, scatter plots of co-ordinates (r and z) of core
samples were made of the 1993 data, with individual points
classified by magnitude of the RLD (Fig. 1). These plots
provided a useful way of projecting 3D data in two
dimensions, hence a method for initial visual examination of
RLD distribution patterns. It is evident from Fig. 1 that for
Mark rootstock under both tree spacings RLD decreased
with increasing depth (z), and horizontal radial distance (r)
within the sampled soil volume. This implied the root
0.0
–0·2
–0·4
0·20
0·20
0·10
–0·6
–0·8
–1·0
0·0
0·10
0·05
0·025
0·05
0·025
Tree = 2, Mark, 5 × 3 m
Tree = 1, Mark, 5 × 3 m
Distance from soil surface (m)
–0·2
–0·4
0·10
–0·6
0.10
–0·8
–1·0
0·0
0.20
0·05
0·025
0.05
0.025
Tree = 3, Mark, 4 × 2 m
Tree = 4, Mark, 4 × 2 m
Tree = 5, MM106, 5 × 3 m
Tree = 6, MM106, 5 × 3 m
–0·2
–0·4
–0·6
–0·8
–1·0
0·0
–0·2
–0·4
–0·6
–0·8
–1·0
0·0
Tree = 8, M26, 4 × 2 m
Tree = 7, M26, 4 × 2 m
0·5
1·0
1·5
2·0
Radial distance from tree (m)
2·5
0·0
0·5
RLD = 0
0 < RLD < = 0·1
0·1 < RLD < = 0·2
0·2 < RLD
1·0
1·5
2·0
Radial distance from tree (m)
2·5
3·0
F. 1. Scatter plots of depth against radial distance, with the magnitude of RLD value indicated by different symbols, for eight apple tree root
systems sampled by soil coring in 1993. The contours shown for Mark rootstock only, are the fitted curves of the Bernoulli–exponential model
(see text).
338
De SilŠa et al.—Analysing Root Length Density of Apple Trees
T     2. Estimates of occupancy (proportion), and the mean root length density of occupied core sample units (RLD+) for
eight apple trees sampled by soil coring in spring 1993
Inner zone*
Outer zone
Tree
Treatment
Sample
size
Occupancy
Mean RLD+
(cm cm−$)
Sample
size
Occupancy
Mean RLD+
(cm cm−$)
1
2
3
4
5
6
7
8
Mark, 5i3
Mark, 5i3
Mark, 4i2
Mark, 4i2
MM.106, 5i3
MM.106, 5i3
M.26, 4i2
M.26, 4i2
28
24
16
20
40
40
21
20
0n61
0n71
0n69
0n90
0n83
0n90
0n90
0n95
0n16
0n11
0n14
0n15
0n13
0n14
0n13
0n17
63
60
27
21
45
41
20
22
0n38
0n25
0n30
0n48
0n60
0n68
0n75
0n82
0n06
0n05
0n02
0n07
0n11
0n12
0n10
0n08
*For Mark, the inner zone is bounded by a semi-ellipse with horizontal and vertical axes of 1n6 and 0n8 m, and for MM.106 and M.26 the inner
zone is defined by a vertical depth 0n5 m.
1·0
Occupancy (proportion)
0·8
Tree No.
1–2: Mark, 5 × 3
3–4: Mark, 4 × 2
5–6: MM.106, 5 × 3
7–8: M.26, 4 × 2
8
7
4
6
5
8
7
2
0·6
6
3
1
5
4
0·4
1
Inner zone
Outer zone
All data
Mark only
3
2
0·2
0
0·05
0·10
0·15
0·20
Mean RLD+ (cm cm–3)
F. 2. Plot of ‘ occupancy ’ (proportion of soil core samples with RLD 0) against mean RLD of non-zero core samples.
system occupied a bowl-shaped volume of soil, similar to
that suggested by Gandar and Hughes (1988) for immature
kiwifruit vines. In contrast, both MM.106 and M.26 showed
a more layered distribution pattern within the sampled soil
volume (Fig. 1). Also, the decline in RLD with depth was
not as strong as for the Mark rootstock (Fig. 1). Furthermore
it appears from Fig. 1 that RLD did not fall as much with
radial distance, particularly in M.26. For the 5-year-old
trees, both MM.106 and M.26 tree roots seem to have
penetrated a greater soil volume than Mark.
In relation to soil coring data, Gandar and Hughes (1988)
defined the term ‘ occupancy ratio ’ as the proportion of core
samples with a RLD greater than a specified value.
Obviously it is a parameter that will depend on the physical
size of the core sample, but provided the same size is used
comparisons can be made among trees belonging to different
treatment groups. In this paper we shall call the ‘ occupancy
ratio ’ at RLD 0 simply ‘ occupancy ’. For each tree
sampled in 1993 we calculated the occupancy for an inner
and outer zone. In the case of MM.106 and M.26, which
exhibited a more layered structure, zones were demarcated
at a depth of 0n5 m. For Mark the inner zone was defined as
bounded by a semi-ellipse with horizontal and vertical axes
of 1n6 m and 0n8 m, respectively. These values were set
somewhat arbitrarily to ensure a reasonable spread of nonzero data across the two zones. A method for estimating the
shape of root volume will be discussed below. Occupancy
differed markedly between the two zones for apple trees on
Mark rootstock (Table 2). On average, occupancy in the
inner zone was 71 % compared to 33 % in the outer zone.
The difference was much smaller for MM.106 (86 % Šs.
64 %) and M.26 (93 % Šs. 79 %).
We plotted the estimated occupancy against the mean
RLD of non-zero sample cores (R- LD+, Table 2). The
resulting plot (Fig. 2) suggests an increasing relationship.
An asymptotic exponential curve
Occupancy l 1kEXP (kb R- LD+)
where the rate constant b is determined from the data, was
fitted. The model fitted the data well (P 0n05, R# l 0n65,
De SilŠa et al.—Analysing Root Length Density of Apple Trees
339
3·5
Mark, 5 × 3 m
3·0
Mark, 4 × 2 m
Radial distance
2·5
0·75 m
1·5 m (between row)
2·0
Mean root length density (cm cm–3)
1·5 m (within row)
1·5
2·25 m
1·0
0·5
0·0
MM.106, 5 × 3 m
3·0
M.26, 4 × 2 m
2·5
2·0
1·5
1·0
0·5
0·0
0·2
0·4
0·6
Depth (m)
0·8
0·0
0·2
0·4
0·6
Depth (m)
0·8
1·0
F. 3. Plot of mean root length density (RLD) against depth, for apple trees belonging to four treatment groups sampled in 1997. Error bars
representp1 s.e.
Fig. 2). A plausible explanation for this relationship can be
based on the assumption that in younger trees fine roots
develop first by a process of random initiation on the
structural root system. This is followed by growth at the
initiation point. Both random initiation and the deterministic growth that follows occur over time during the
season. Therefore, the higher the occupancy the greater the
root length density at initiated points. If data captured by
coring are a realization of this process, then the occupancy
could be considered to reflect the level of initiation, and
R- LD+ the growth. This may be a plausible conceptual
model for young trees with sparse RLD. For older trees at
a more advanced stage of root development the process is
likely to be complicated by simultaneous death of fine roots.
We will make use of the asymptotic exponential relationship
of Fig. 2 for the purposes of developing a model for root
coring data in a later section. Fitted values of the rate
parameter were 11n2 for all the data, and 8n9 for Mark
rootstock only (Fig. 2).
As described earlier, the 1997 sampling was carried out on
a rectangular horizontal grid of 1i1 m squares, placed with
the tree at the centre. Samples along the soil core were also
taken equi-distantly. Hence scatter plots of (r, z) coordinates as in Fig. 1 were not produced due to clustering of
r values. Instead, we plotted mean RLD (R- LD) against
depth for the three radial distances at which cores were
sampled (Fig. 3). These were approx. 0n75, 1n5 and 2n25 m
T     3. Estimates of occupancy (proportion), and the mean
root length density of occupied core sample units (RLD+) for
eight apple trees sampled by soil coring in spring 1997
Tree
1
2
3
4
5
6
7
8
Treatment
group
Occupancy
Mean RLD+
(cm cm−$)
Mark, 5i3
Mark, 5i3
Mark, 4i2
Mark, 4i2
MM.106, 5i3
MM.106, 5i3
M.26, 4i2
M.26, 4i2
0n93
0n94
0n95
0n90
0n95
0n96
1n00
1n00
0n69
0n82
0n80
0n62
0n48
0n74
0n72
0n83
radially distant from the base of tree trunk. For the 5i3 m
spacing, the cores at approx. 1n5 m radial distance were at
angles of either about 18m (within-row) or 72m (between-row)
from the row direction. Between 1993–1997, the mean RLD
of the apple root systems increased almost ten-fold. It is also
evident (Fig. 3) that for older trees the R- LD increased with
radial distance within the surface area sampled. This is quite
the opposite of observations for younger trees. This trend
was consistent across all four treatment groups. A further
340
De SilŠa et al.—Analysing Root Length Density of Apple Trees
interesting observation in the 1997 data was that for the
same radial distance (1n5 m), cores taken from between rows
(72m from the row direction) had a higher density compared
with those sampled from within the row (18m) (Fig. 3). As
expected, the RLD declined exponentially with increasing
depth. In terms of overall tree mean RLD there appears to
be no differences in occupancy between treatment groups
(Table 3), with occupancy values exceeding 90 % for all
trees.
REGRESSION MODEL ANALYSIS
The objective of the regression model analysis was to
ascertain if treatment groups differed significantly in their
R- LD at different radial distances from the tree and at
varying soil depths. The 1993 RLD data were sparse with a
high proportion of zeros, hence, prior to model fitting, R- LD
values were calculated for combinations of treatmentgroupiradial-distanceidepth, ignoring any tree effects.
Although some differences between trees within a group
were evident (Fig. 1) as discussed previously, averaging over
the two replicate trees was expected to provide more robust
mean values. For the purposes of this analysis, the radial
distance of cores sampled in 1993 were classified as follows :
0n6 : 0n3–0n9 m ; 1n2 : 0n9–1n5 m ; 1n8 : 1n4–2n1 m ; 2n4 : 2n1 m.
Similarly, the depths of core samples were classified as : 0n15,
0n30, 0n45, 0n60, 0n75 and 0n90 m, with each level including
0·25
Mark, 5 × 3 m
0·20
Mean root length density (cm cm–3)
0·15
0·10
depths of p0n075 m. The 2n4 m radial distance category
was excluded from analysis to provide a more balanced data
set across the two tree-spacings.
For the 1993 data, we defined a general model as :
yijk l log (R- LD) l µjαijλjjαλijkβij xkjεijk
where R- LD is the mean RLD of the i-th treatment group,
at the j-th radial distance from the tree, and at a depth of
xk m. The model assumed that R- LD changed proportionately with the level of radial distance, but the inclusion of
the interaction term, αλij, implied the drop may be treatment
group dependent. The R- LD was assumed to drop exponentially with depth, xk, which is included as a covariate
in the model. According to the general model above, we
assume this rate of decline (βij) to be both treatment group
and radial distance dependent. Parsimonious versions of the
model can be fitted to test for any non-specificity in the rate
parameter.
In terms of the regression model, another aspect needing
consideration is the assumption of independence of errors,
εijk, within a ij-th group. Since the mean values at different
depths (for a given radial distance) are based on the same set
of cores these errors may be correlated, i.e. if ε is a vector of
within-subject (core type) residuals, then E(ε) l 0 but
Cov(ε) σ#I, where I denotes an identity matrix. Instead,
the subjects produce a block diagonal structure in R,
[Cov(εijk)] with identical blocks. Since in eqn (1) the εijk
Mark, 4 × 2 m
Radial distance
0·6 m
1·2 m
1·8 m
0·6 m
1·2 m
1·8 m
0·05
0·00
MM.106, 5 × 3 m
M.26, 4 × 2 m
0·20
0·15
0·10
0·05
0
0·2
0·4
0·6
Depth (m)
0·8
(1)
0·0
0·2
0·4
0·6
Depth (m)
F. 4. Fitted curves of regression model [eqn (1) in text] to 1993 mean RLD data.
0·8
1·0
De SilŠa et al.—Analysing Root Length Density of Apple Trees
341
T     4. Regression model estimates of mean surface (at zero depth) RLD at Šarious radial distances from the tree in two
seasons for apple trees belonging to different rootstocks and spacing treatment groups
Radial*
distance
(m)
Mark, 5i3
Mark, 4i2
MM.106, 5i3
M.26, 4i2
Estimate (s.e.r.†)
Estimate (s.e.r.)
Estimate (s.e.r.)
Estimate (s.e.r.)
0n67 (1n75)
0n03 (2n21)
0n01 (1n73)
0n14 (1n73)
0n17 (1n68)
0n10 (1n55)
0n18 (1n81)
0n12 (1n84)
0n30 (1n92)
0n78 (1n27)
0n89 (1n25)
0n82 (1n27)
1n77 (1n25)
3n25 (1n27)
0n95 (1n27)
1993 root sampling
0n6
0n22 (1n67)
1n2
0n16 (1n52)
1n8
0n05 (1n62)
1997 root sampling
0n75
0n80 (1n25)
1n50 wr
1n25 (1n27)
1n50 br
2n64 (1n25)
2n25
4n06 (1n27)
3n56 (1n27)
1n69 (1n27)
* wr, Within-row, 18m from row direction ; br, between-row, 72m from row direction.
† Standard error ratio : divide and multiply the estimate by s.e.r. to get the lower and upper bounds corresponding to one standard error.
terms are assumed to be normal, the corresponding error
terms of yijk are assumed to be log-normal distributed.
Unlike simple linear models, mixed models allow for random
effects as well as both correlation and non-identicality of
error effects. First, we fitted the mixed model by REML
(restricted maximum likelihood) using SAS statistical
software (SAS, 1996) to test for non-independence of errors.
Secondly, if the correlation was not significant the model
was fitted assuming independence of errors.
Results of 1993 analysis
The mixed model analysis of 1993 data provided no
evidence of correlation between depths within a soil core
type. The likelihood ratio test for the independent constantvariance null model against the general Toeplitz covariance
structure (same variance parameter for diagonals, but off
diagonal elements depend on level of separation) was not
significant [k2∆ log (likelihood) l 4n9, χ#, P 0n05]. The
&
estimated Pearson correlation coefficients were in the range
0n16 to 0n26.
The model described by eqn (1) was then fitted by the
method of weighted least-squares, with weights equal to the
number of data points per R- LD observation. In spite of the
relatively large amount of noise in the data, the fit of eqn (1)
was reasonably good (R# l 0n77, Fig. 4). The two main
effects, their interaction and the covariate were all significant
(P 0n01). Table 4 gives the fitted R- LD at zero depth. We
shall refer to this, which is the hypothetical intercept on the
y-axis (Fig. 4), as the surface R- LD. It is a measure of the
magnitude of R- LD at a point on the surface. From there
onwards the model indicated that R- LD declines exponentially with depth. We fitted more parsimonious forms of
eqn (1) and tested whether tree-spacing and rootstock
affected the RLD profile. When compared with Mark at the
same spacing, both MM.106 (8n1, " F , , P 0n01) and
' %)
M.26 (5n4, F , , P 0n01) exhibited significantly different
' %)
RLD profiles. In contrast the difference between Mark at
the two spacings was less conspicuous (3n1, F , , P 0n05).
' %)
The differences are highlighted by the results presented in
Table 4. For Mark, the surface R- LD declined sharply with
radial distance, particularly at 1n8 m from the tree. In
contrast, for MM.106 and M.26, the mean surface RLD
did not differ much with radial distance within the sampled
area. This implied that both MM.106 and M.26 had more
uniform horizontal distributions than Mark. Differences in
the rate of decline of RLD with depth were also evident. We
shall examine the rate coefficient only for the shortest radial
distance (0n6 m) from the tree. For Mark, RLD values
beyond this were too low to make any meaningful
comparisons. The fitted slope parameters at a distance of
0n6 m (Fig. 4) were 3n04p0n71, 4n58p0n96, 0n84p0n95 and
2n01p1n06, respectively, for Mark 5i3, Mark 4i2,
MM.106 5i3 and M.26 4i2 treatment groups. These
estimates suggest that for both MM.106 and M.26 the mean
RLD declined with depth at a lower rate than Mark.
The greater surface RLD and a lower rate of decline with
depth meant that MM.106 rootstock had a significantly
greater (P 0n05) mean RLD at 0n5 m depth, for all three
radial distances, compared with Mark at the same treespacing. Similarly, M.26 had significantly greater (P 0n01)
R- LD values compared with Mark, at 1n2 and 1n8 m radial
distances, but no difference at the 0n6 m radial distance. The
difference between the two spacings within Mark rootstock
was not convincing, with a significant difference detected
only for the 1n2 m radial distance.
Results of 1997 analysis
As with the preceding analysis, a mixed model was fitted
to the 1997 data to test if the observed mean RLD values
within a core were correlated. Again, there was no evidence
of any association [k2∆ log (likelihood) l 3n5, χ#, P 0n05],
%
with estimated correlation coefficients ranging up to 0n32.
The model described by eqn (1) was therefore fitted assuming
independence of errors and constant variance. Since the
treatment groups were non-orthogonal in respect of the
different levels of radial distance, the term αλij was included
in the model without the corresponding terms for main
effects. A more parsimonious version of this model gave the
342
De SilŠa et al.—Analysing Root Length Density of Apple Trees
3·0
Mark, 5 × 3 m
Mark, 4 × 2 m
0·75 m
1·5 m (between row)
1·5 m (within row)
2·25 m
2·5
2·0
Mean root length density (cm cm–3)
1·5
1·0
0·5
0·0
MM.106, 5 × 3 m
M.26, 4 × 2 m
2·5
2·0
1·5
1·0
0·5
0·0
0·2
0·4
0·6
Depth (m)
0·8
0·0
0·2
0·4
0·6
Depth (m)
0·8
1·0
F. 5. Fitted curves of regression model [eqn (1) in text] to 1997 mean RLD data.
best fit to the data, with an R# value of 0n91. The rate
coefficient βij dependent only on the treatment group (P
0n01) and the radial distance (P 0n05), and not on the
interaction of the two. The αλij effects were highly significant
(P 0n01). The fitted curves are given in Fig. 5, and the
fitted values of mean surface RLD are given in Table 4. As
before, we fitted more parsimonious models and tested the
significance of spacing and rootstock effects on the RLD
profile. Spacing did not significantly affect the RLD profile
of Mark rootstock (1n1, " F , , P 0n05). The RLD profile
$ %"
of MM.106 was not significantly different (1n3, " F , , P & %"
0n05) from Mark at the same spacing. The difference
between Mark and M.26 was highly significant (14n4,
" F , , P 0n01). At the same radial distance of 1n5 m, the
$ %"
RLD profile of soil cores sampled between-rows differed
significantly (P 0n05) from those sampled within-rows
(Fig. 5), with the former showing a higher R- LD. Within a
treatment group, the surface R- LD showed a consistent and
systematic trend with radial distance (Table 4). The RLD
increased with increasing distance from the tree trunk, with
the values at 2n25 m radial distance being as much as 5n1and 3n7-times greater for Mark and MM.106, respectively,
at the 5i3 spacing, compared with those at 0n75 m. These
results are the opposite to those reported earlier for younger
trees sampled in 1993.
The R- LD declined exponentially with depth and the rate
of decline varied with radial distance. The higher surface
R- LD values further away from the tree trunk dropped off at
a faster rate compared with lower values closer to the tree
trunk. For treatment group 1 (Mark, 5i3) the estimated
rate coefficients were 1n98, 2n17, 2n58 and 3n27, respectively,
for distances of 0n75, 1n5 (within-row, 18m), 1n5 (between
row, 72m) and 2n25 m (Fig. 5) from the tree trunk. The
significant differences in RLD profile between Mark and
M.26 are reflected in their fitted values of surface R- LD and
the rate coefficient. The R- LD of M.26 declined with depth
at a slower rate than Mark (0n81, 1n41 relative to 2n46,
3n06 at 0n75 and 1n5 m radial distance). At the 0n75 m
distance, M.26 had a higher surface R- LD than Mark, but
the trend was reversed at 1n5 m (Table 4).
A MIXTURE MODEL
Model formulation
In comparison with the ANOVA model [eqn (1)], a better
approach to modelling the RLD data is to fit some empirical
density function. The core sample data can be viewed as
being generated from a Bernoulli-exponential process. We
assume the presence of roots in each core section is
independent of others within the same core, and any other
disjoint core section in any given region of root space. Each
core section has a fixed probability p (occupancy) of
presence of roots. If roots are present in a core section, then
RLD is a continuous variable and assumed to be exponentially distributed. The population distribution func-
De SilŠa et al.—Analysing Root Length Density of Apple Trees
tion of the random variable of RLD, D, can therefore be
written as :
1
dl0
FD(d ) l
1kp
2
&! β1 e
d
3
1kpjp
4
−t
β
dt d 0
(2)
A soil coring survey of kiwifruit root systems by Gandar
and Hughes (1988) showed that for younger vines mean
RLD fell with depth and radial distance from the vine.
Hence, young root systems are generally bowl-shaped, and
regions with similar RLD values are likely to appear as
concentric contours. Gandar and Hughes (1988) defined
these contours in terms of concentric ellipses. The contour
plots of Fig. 1 suggest a similar bowl-shaped structure for
apple root systems of Mark rootstock. The root systems of
MM.106 and M.26 appear to be more layered than bowlshaped. The general equation for an ellipse is :
(4)
where r is radial distance, z is depth, and A and B are
constants. To describe the bowl-shaped root system we
constrained eqn (4) to a family of concentric ellipses by
writing B as a function of A in the form, B l c NA, where
A is a measure of the distance of an ellipse from the origin,
and c is a constant for a given family of curves. This squareroot relationship implies that ellipses become flatter as the
distance from origin increases. Rearrangement gives A as a
function of r, z and c :
Al
9 ’
z#
1j
2c#
1j4
:
r# c%
z%
likelihood of a set of data with m zero values and (nkm)
non-zero values is given by :
L(β , κ , κ , c ; [r , z , d ], … [rm, zm dm], … [rn, zn, dn])
! " #
" " "
where β is the parameter of the exponential density function
and is equal to the E(D+). If we consider the initialization of
fine roots on the root system and their subsequent growth as
processes that progress in time within a given season, then
it is plausible to assume that larger RLD+ values are
associated with higher occupancies, i.e. p l f(E(D+.)). We
shall assume a simple asymptotic exponential for this
relationship :
(3)
p l 1ke−κ"β
r# z#
j l1
A # B#
343
(5)
The above mathematical description of geometry of the root
system means that we can now express β in eqn (3), which
is the E(D+), as a function of the distance from origin. Here
we shall assume β declines exponentially with distance :
β l β e−κ# A
(6)
!
where β is the value of β at the origin (A l 0), and κ is a
!
#
rate constant. Now, we have the original distribution
function [eqn (2)] given in terms of four parameters : β , κ ,
! "
κ and c.
#
Estimation of model parameters
Model parameters are estimated using the method of
maximum likelihood. From eqn (2) it follows that the
m
n 1 dj
e− βj
l (1kpj) β
j="
j=m+" j
(7)
where pj and βj are given by eqns (3) and (6), respectively.
The log-likelihood, l, is :
9
m
n
d
l l log (1kpj)k log (βj)j j
β
j
j="
j=m+"
:
(8)
We defined the function l in the Interactive Matrix Language
(IML) of SAS (SAS, 1996). The four parameters were then
estimated by maximizing the objective function using the
SAS\IML Newton-Raphson optimization subroutines. The
initial values for θ l (β , κ , κ , c) were set to (0n2, 7, 0n3, 0n4).
! " #
A relative gradient convergence criterion was used, with
termination requiring that the normalized predicted function
reduction be small ( 1Ek8). Approximate standard errors
for parameter estimates were obtained from the diagonal
elements of the inverse of Hessian, evaluated at the optimal
parameter estimates. The Hessian was calculated by finite
differencing using a SAS\IML algorithm. The optimizations
converged easily, taking only a few iterations in each
instance. When the model was fitted to several treatment
groups, invariance of parameters between groups was tested
using the likelihood ratio tests (LRT). The test statistic,
which is k2 times the difference in log likelihood is
approximated by the χ# distribution with degrees of freedom
equalling the difference in number of parameters between
the given model and its parsimonious derivative.
Application to apple data
Since a bowl-shaped symmetry in the root system was
only apparent for trees of Mark rootstock (Fig. 1), the
Bernoulli-exponential model was fitted only to this subset of
data, which contained two spacing treatment groups, each
with two replicate trees. The most general model fitted made
the parameter β tree dependent and all others tree-spacing
!
dependent, a total of ten parameters. Estimates of maximum
likelihood for the general and various sub-models are given
in Table 5. According to these results, the only significant
decrease in log-likelihood, as indicated by the LRT (8n22, χ#,
#
P 0n05), was when tree dependence of β was dropped
!
from the model. There was no evidence that any of the four
parameters were dependent on tree-spacing : i.e. the distribution of RLD was not affected by spacing of apple trees
on Mark rootstock. This result is consistent with that
obtained with the regression model approach. The estimates
of parameters (β I , β I , β I , β I , κ , κ , c) for the optimal
!" !# !$ !% " #
model [model (8) of Table 5] were 0n29, 0n25, 0n20, 0n36, 8n8,
0n73, 0n69, where β I , … β I , defined the parameter β for
!%
!
!"
trees 1 to 4. The corresponding s.e. estimates were 0n059,
0n052, 0n043, 0n077, 1n16, 0n113 and 0n078, respectively. Fitted
contours of the Bernoulli-exponential model for overall
R- LD values (piD̀+), of 0n2, 0n1, 0n05 and 0n025 cm cm−$ are
344
De SilŠa et al.—Analysing Root Length Density of Apple Trees
T     5. Summary of Bernoulli–exponential model fitted to root core data from four apple trees (Mark rootstock at two
spacings, and two replicates) to test the inŠariance of model parameters
Model
No. of
parameters
Loglikelihood
10
17n77
9
9
9
8
8
8
7
5
4
16n42
17n15
17n73
16n38
17n00
15n92
15n80
11n69
11n64
0. Tree dependent β ,
!
spacing dependent κ , κ , and c
" #
1. Invariant κ
"
2. Invariant κ
#
3. Invariant c
4. Invariant κ and c
"
5. Invariant κ and c
#
6. Invariant κ and κ
"
#
7. Invariant κ , κ and c
" #
8. Spacing dependent β
!
9. Common β , κ , κ and c
!
"
#
k2∆ ln (L)†
2n70 (1)
1n24 (1)
0n08 (1)
2n78 (2)
1n54 (2)
3n70 (2)
3n94 (3)
12n16 (5)*
12n26 (6)
† Against the general model (0) ; difference in degrees of freedom is within brackets.
* P 0n05.
shown in Fig. 1. For trees on Mark rootstock, these
contours describe the scatter-plot data reasonably well.
DISCUSSION
We have described and compared the RLD distribution of
apple trees belonging to three different rootstocks, in the
same orchard block in the fifth and ninth year of planting.
The dwarfing effect of the rootstocks used in this study,
from greatest to smallest, are : Mark (dwarf), M.26 (semidwarf) and MM.106 (semi-vigorous). The RLD, given as cm
of fine roots per cm$ of soil volume, was extremely low when
trees were young. Estimates of overall mean RLD, to a
depth of 1 m, ranged from 0n03–0n08 cm cm−$ for trees of
Mark rootstock, and 0n09–0n11 cm cm−$ for others. These
values compare well with the 0n1–0n2 cm cm−$ range obtained
by Hughes and Gandar (1993) for 4-year-old non-dwarf
apple trees. Both results further confirm that apple trees
have a very sparse root system (Atkinson, 1980). This study
also demonstrated that the RLD of apple trees increased
almost ten-fold over a 4 year period of growth (overall mean
from 0n07 to 0n67 cm cm−$). In contrast, Hughes and Gandar
(1993) reported that RLD of apple trees reached its
maximum at about 4 years of planting. One aspect that has
not been considered carefully here is the timing of sampling
within a given season. Both the 1993 and 1997 core
samplings were carried out in spring, with the latter about
a week later in the calendar year. It is known that in fruit
trees, flushes of new roots occur in spring (Atkinson, 1980).
Exact timing of sampling, especially in relation to aboveground shoot growth, may be important for studies of
temporal change over seasons. This study has provided
clear evidence of a striking difference in the horizontal RLD
distribution between young and older trees. The root
systems of young trees, particularly that of Mark rootstock,
exhibited a bowl-shaped structure with RLD decreasing
with increasing radial distance from the tree trunk. In
contrast, the RLD of older trees increased with distance
within the sampled area. The result is not unexpected
because these trees lack a tap root and the lateral structural
roots grow and branch outwards providing, over time, a
greater root biomass away from the tree trunk. Irrigation
can also affect root growth (Neilsen et al., 1997). The
experimental plots of this study were uniformly irrigated
using mini-sprinklers located along the tree line. These
sprinklers emitted water up to 1 m radius around the tree,
hence irrigation is unlikely to have caused the high RLD
2 m from the tree trunk. In older trees there was also
evidence that at a given radial distance, the between-row
area had a higher RLD than the area within a row. It would
be interesting to investigate this further because it may have
important implications for how the trees are irrigated.
This study demonstrated that for young apple trees, the
mean RLD and its distribution differed significantly between
different rootstocks. Both the M.26 and MM.106 rootstocks
had a higher mean RLD and a more layered distribution
compared with Mark at the same tree-spacing. When trees
were sampled 4 years later, however, there were no
significant differences in the tree mean RLD among
rootstocks. The only difference observed in RLD profile in
older trees was that M.26 showed a more layered vertical
distribution than the Mark rootstock. In neither season was
there evidence of a significant spacing effect on the RLD
profile.
In terms of statistical analysis of RLD data in this study,
we used the standard weighted least squares method to fit a
regression model to log transformed data. The residual
plots did not indicate any functional relationship between
the mean and variance. Although a generalized linear model
analysis may have provided a better justification of model
assumptions, the resulting test statistics would have only
approximate distributions. We believe our approach to the
analysis was reasonable because the model was fitted after
averaging which resulted in very few zeros compared with
the raw data. In this paper we have also successfully
developed and illustrated the use of a model which assumed
the realization of RLD data was due to two random
processes : a discrete Bernoulli and a continuous exponential.
The model provided a more biologically meaningful
description of the RLD distribution. The estimation of
model parameters was carried out by maximum likelihood,
and the iterations converged rapidly. Although the
De SilŠa et al.—Analysing Root Length Density of Apple Trees
Bernoulli-exponential model has been fitted only to the
bowl-shaped root system, this approach could be generalized
to root systems with different geometries.
A C K N O W L E D G E M E N TS
Mr K. Hughes for conducting the 1993 sampling study, and
for useful comments for the later study. Messrs A. K. N.
Zoysa, C. Van Den Dijssel, J. J. Tienstra, J. F. Julian, B. J.
Jardine, and Ms Karina Foote for assistance with soil
coring and data collection.
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