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Ann Bot. 2011 November; 108(7): 1279–1286.
Published online 2011 September 13. doi:  10.1093/aob/mcr228
PMCID: PMC3197456

Common allometric response of open-grown leader shoots to tree height in co-occurring deciduous broadleaved trees


Background and Aims

Morphology of crown shoots changes with tree height. The height of forest trees is usually correlated with the light environment and this makes it difficult to separate the effects of tree size and of light conditions on the morphological plasticity of crown shoots. This paper addresses the tree-height dependence of shoot traits under full-light conditions where a tree crown is not shaded by other crowns.


Focus is given to relationships between tree height and top-shoot traits, which include the shoot's leaf-blades and non-leafy mass, its total leaf-blade area and the length and basal diameter of the shoot's stem. We examine the allometric characteristics of open-grown current-year leader shoots at the tops of forest tree crowns up to 24 m high and quantify their responses to tree height in 13 co-occurring deciduous hardwood species in a cool-temperate forest in northern Japan.

Key Results

Dry mass allocated to leaf blades in a leader shoot increased with tree height in all 13 species. Specific leaf area decreased with tree height. Stem basal area was almost proportional to total leaf area in a leader shoot, where the proportionality constant did not depend on tree height, irrespective of species. Stem length for a given stem diameter decreased with tree height.


In the 13 species observed, height-dependent changes in allometry of leader shoots were convergent. This finding suggests that there is a common functional constraint in tree-height development. Under full-light conditions, leader shoots of tall trees naturally experience more severe water stress than those of short trees. We hypothesize that the height dependence of shoot allometry detected reflects an integrated response to height-associated water stress, which contributes to successful crown expansion and height gain.

Keywords: Allometry, current-year leader shoot, hierarchical Bayesian model, pipe model, tree height, water stress


The above-ground branching architecture of trees is composed of numerous morphological components that vary in size, structure and function. A current-year shoot, as the unit of annual shoot growth in deciduous trees, is an assemblage of functional components, i.e. leaf blades, petioles and stem (usually un-branched). Development of the crown is the consequence of the dynamics of the current-year shoot population on the previous-years' populations (Room et al., 1994). Deciduous trees that renew their leaves on current-year shoots each year expand foliage clusters by lateral-branch growth and vertically by main trunk growth. This enables trees to compete for light with their neighbouring trees. At the scale of a whole tree, height gain is associated with an investment in a thickening of the trunk and lower older branches to maintain mechanical stability (McMahon, 1975; King, 1986) and to transport water (Shinozaki et al., 1964; Whitehead et al., 1984). Consequently, the cost of radial growth of aged stems inevitably increases with tree height. To maintain the amount of tree-level assimilation area, each current-year shoot must increase the mass allocation of assimilate to leaf blades relative to non-leafy parts (stem and petioles). On the other hand, a decrease in leaf area relative to the sapwood area with tree height can compensate for increasing resistance to water transport in tall trees (McDowell et al., 2002). These aspects at the scale of a whole tree are expected to influence the morphological traits of current-year shoots, such as mass allocation between leaf blades and non-leafy parts, total leaf area, stem length and stem basal diameter.

To examine these morphological responses of shoots to tree height, attention must be paid to the ambient environment of a shoot, such as light conditions which change with stand-level canopy structure, crown height and the position of the shoot within a crown. Because short trees usually grow under dark conditions in forests and because the light conditions of shoots change within a crown due to self-shading (Sterck and Bongers, 2001; Ishii et al., 2008), the response of a whole current-year shoot to tree height independent of light conditions has not previously been determined. In contrast to a whole current-year shoot, the leaf-level response to tree height under full-light conditions so far examined shows a general trend for the specific leaf area (SLA) of fully exposed leaves to decrease with tree height (Nabeshima and Hiura, 2008; Ambrose et al., 2009; Woodruff et al., 2009). The height-related decline in leaf turgor due to a decrease in water potential with both hydraulic resistance and gravity is a key factor leading to a reduction in foliar expansion and a consequent reduction in SLA (Woodruff et al., 2008; Mullin et al., 2009). We pose here a question for the mechanical understanding of shoot-level morphological response to tree height. To answer this, it is crucial to focus on the relationship between leaf and stem because it is the performance of a whole shoot that contributes to the growth of the tree. We need to examine the response of a current-year shoot to tree height separately from its response to light. A simulation model suggests that the morphology of current-year shoots changes with increasing tree height, if the pipe model assumption in which the stem basal area is proportional to the total leaf area (Shinozaki et al., 1964) is an important constraint in tree development (Kubo and Kohyama, 2005).

For further mechanical understanding of the plastic morphology of current-year shoots, this paper offers five predictions: (1) SLA decreases with tree height (cf. Nabeshima and Hiura, 2008), (2) the proportion of stem basal area to total leaf area is kept unchanged over a range of tree height (cf. Shinozaki et al., 1964), (3) the leaf-blade mass supported by a given stem basal area increases with tree height (based on predictions 1 and 2) and (4) the leaf-blade mass supported by a given mass of non-leafy parts increases with tree height (cf. Kubo and Kohyama, 2005). Therefore, we can also predict (from 1–4) that (5) the stem length of the shoot relative to the stem basal diameter decreases with tree height. As far as we are aware, there have been no studies of shoot-scale responses to tree height under exposed light conditions across several species in the same habitat. To examine these questions and predictions, we observed a current-year leader shoot at the top of the open-grown crown of 13 co-occurring deciduous hardwood canopy/subcanopy species of various tree heights. We expect that the height-dependent shoot response is common across tree species because of the common demand for canopy/subcanopy trees to increase their height. The conventional method of allometry analysis fits power functions (log-linear functions) between two size dimensions, thereby ignoring other interdependent dimensions (Price et al., 2009). To avoid this problem and to evaluate height-dependent change in shoot morphology across species, we employed Bayesian statistical techniques (Clark, 2005; Price et al., 2009), which allowed us to untangle cross-species similarity and species-specific responses across an interdependent set of allometric functions.


Study site and species

Leader shoots were sampled in a cool-temperate, deciduous broadleaved forest in the Tomakomai Experimental Forest (TOEF) (42°40′N, 141°36′E) of Hokkaido University, northern Japan. Mean monthly temperature ranges from −3·2 to 19·1 °C, and annual precipitation is 1450 mm, the peak of which occurs in summer (Hiura, 2005). Located on the foothill plateau of Mt Tarumae (1041 m), an active volcano, this site has an altitude range from 5 to 90 m with gentle sloping topography and the water table is constant across the topography and across seasons (Shibata et al., 1998). This forest has been established on a volcanic ash deposit of approximately 2 m depth formed by two eruptions of Mt Tarumae in 1669 and 1739. The surface of the volcanic substrate is covered by organic soil to a depth of about 50 cm (Igarashi, 1987). Abundant species in TOEF are Acer mono, A. amoenum, Ostrya japonica and Quercus crispula. Details of the vegetation and the soil are reported in Shibata et al. (2001) and Hiura (2005). We chose 13 abundant deciduous tree species as listed in Table 1. Among them, canopy species grow up to 26 m whereas sub-canopy species grow up to a maximum of 20 m (Hiura et al., 1998).

Table 1.
Species, species codes and sample sizes of current-year leader shoots examined

Sampling and measurement of morphological traits

In July and August 2006, we selected sample trees for each species with heights ranging from 1 to 24 m over the study site. We sampled an uppermost shoot, located at the top of each open-grown tree crown, which was not affected by mutual shading among its own shoots nor by neighbouring crowns. We chose leader shoots without significant damage from herbivores, and measured the vertical location of each shoot from the ground. Tall trees (16−24 m) were chosen around a canopy crane (25 m high, 41·5 m jib), which was located in a mature forest stand. Using this crane we sampled shoots of eight of 13 species (Table 1). We also selected trees of tall and intermediate heights (8−23 m) growing along sunny roadsides, including five species that were not found around the crane. To sample leader shoots at the tops of trees, a bucket truck with an arm height of 13 m was used. We also took leader shoots from small trees (1−8 m) that grew in large forest gaps and along sunny roadsides using pruning shears.

We chose 494 leader shoots (taken from 494 different trees), which were cut at the base of the bud scars of the previous winter. Each leader shoot was divided into leaf blades and non-leafy parts (petioles and stem). Stem length and stem basal diameters (two diameter measures at right angles) were measured. Leaf blades were scanned and digitized to obtain the total leaf area of the shoot. The area of the scanned images was measured with ImageJ software (ver.1·37, We rated insect damage to the leaf area on a five-point scale (0−4) for each shoot (0 for 0−10 % damage, 1 for 10−25 %, 2 for 25−50 %, 3 for 50−75 %, and 4 for 75−100 %). Leaf damage ranged from 0 to 3 with an average of 1·1. All shoot parts were dried at 80 °C for 2 d to measure oven-dry mass. Table 2 shows the ranges of each shoot dimension. Out of 494 leader shoots, we observed only two multi-stemmed current-year shoots with sylleptic branching in which some of the axillary buds had elongated with no dormancy. These were in canopy trees of Magnolia obovata and Betula platyphylla. These exceptional samples were excluded from the analysis.

Table 2.
The observed range of shoot dimensions (minimum-maximum) for 13 deciduous hardwood species studied

Statistical model of allometric relationships

To estimate the effects of tree height on shoot dimensions, we developed a hierarchical Bayesian model in which we dealt with the common characteristics across species and with the differences among species. We quantified the tree-height dependence of total mass, Mi, of leader shoot i (see Supplementary Data – Methods and Table S4, available online). The coefficients of four allometric relationships in shoots were estimated, i.e. allometry 1 between total leaf-blade mass (Fi) and non-leafy part mass (Si), allometry 2 between total leaf-blade area (Ai) and total leaf-blade mass (Fi), allometry 3 between stem basal diameter (Di) and total leaf-blade area (Ai), and allometry 4 between stem length (Li) and stem basal diameter (Di; Fig. 1). In each allometry j, an observed objective variable of a shoot dimension with its shoot-identity i of all species pooled, Yi,j(O), was assumed to follow a normal distribution N with an expected latent variable Yi,j. In other words, latent variables are inferred from observed variables. The equation to assume Yi,j(O) was Yi,j(O) ~ N (Yi,jj(O)), where σj(O) represents the standard deviation of measurement accuracy in Yi,j(O), in which the superscript (o) denotes the observed quantities. We introduced the allometric equation, Yi,j = αi,jXi,jβi,j, where the latent objective variable Yi,j is a power function of the latent explanatory variable Xi,j. We defined the allometric coefficient αi,j to be equal to the linear predictor with the effects of the tree height: αi,j = aj + bjlogHi* + epsiloni,j, where aj is the intercept, bj is the coefficient of dependence on tree height Hi* and epsiloni,j represents the random effect of individual, of shoot i. In the allometry 2 between total leaf-blade area and total leaf-blade mass, βi,j was set to one because we focused on the tree-height dependence of SLA. We set all possible responses in allometry (i.e. in the case of tree height and random effect of each shoot) occurring in αi,j but not in βi,j.

Fig. 1.
Statistical model of five components: total shoot mass and each dimension in a leader shoot i. Circled and boxed symbols show latent variables and observations, respectively. H, tree height; M, total shoot mass; S, mass of non-leafy parts; F, total leaf-blade ...

As we had a set of observed tree heights {Hi(O)}, the latent variable of tree height Hi was modelled such that Hi(O) follows a normal distribution with mean Hi and standard deviation σH(O), Hi(O) ~ N(Hi, σH(O)). The prior of Hi has a non-informative prior distribution defined by Hi ~ N(0,102). A non-informative prior with uniform distribution structure is used when we have no information for target data.

Hierarchical priors for species and individual differences

We introduced hierarchical priors that represent the differences among species and individual trees (or shoots), on the basis of the data. We assumed in our Bayesian model that priors for the parameters defined above have a hierarchical structure as follows. A parameter qj [set membership] {aj, bj, βj} was divided into two components: a common one for all species qj,ALL and a species-specific one, qj,SPC, namely qj = qj,ALL + qj,SPC. The prior of qj,ALL was defined as non-informative, qj,ALL ~ N(0,102), and that of qj,SPC was defined as qj,SPC ~ N(0, σj(q)) where the variation of species differences is controlled by parameter σj(q), which follows a Gamma distribution with mean 1 and variance 102. We also modelled random effect epsiloni,j in αi,j using a hierarchical prior and set a fixed value to the standard deviation σj(O) representing the measurement accuracy in Yi,j(O) to distinguish random effect epsiloni,j. σj(q) and the standard deviation for epsiloni,j, σj(epsilon) follow a Gamma distribution with mean 1 and variance 102. Positing non-leafy part mass and damage scales on the leaf blades is detailed in the Supplementary Data.

Parameter estimation, or the sampling of posterior distributions, was performed using the Markov chain Monte Carlo (MCMC) method with WinBUGS 1·4·3 (Spiegelhalter et al., 2003) and the R2WinBUGS package (Sturtz et al., 2005) on R 2·9·1 (R Development Core Team, 2009). The posterior samples were obtained from three independent Markov chains in which 2400 values were sampled with ten iteration intervals after a burn-in of 2000 iterations. The convergence of the Markov chains was checked with An external file that holds a picture, illustration, etc.
Object name is mcr2281.jpg (Gelman et al., 2003) for each parameter. The An external file that holds a picture, illustration, etc.
Object name is mcr2282.jpg values obtained were less than 1·1 for all parameters. The median and 95 % Bayesian confidence interval or 95 % credible interval (CI) for each parameter were evaluated using the MCMC samples. In the case that the 95 % CI for a parameter included zero, we classified the parameter into the group [no effect]. Otherwise, the effects of parameters were classified into groups of [negative] and [positive] according to the sign of the median of the posterior distributions of each parameter. All source code lists for the analysis were written in R and BUGS languages [Supplementary Data – Coding].

We successfully obtained distributions of model parameters that characterize shoot traits. The medians and 95 % CIs of {aj,bjj} are listed in Supplementary Data Tables S1–S3. We focused on the tree-height dependence across four allometries of shoot traits, namely parameter bj of each allometry j. The estimated effects of tree height were categorized into three groups: ‘common response’ across the species examined, ‘species response’ and ‘species difference’, which correspond to bj,ALL, bj,ALL + bj,SPC and bj,SPC, respectively. Species response to tree height (bj,ALL + bj,SPC) was larger or smaller than that of other species when the species difference (bj,SPC) was positive or negative. We predicted five shoot dimensions, and standardized responses to tree height for each dimension in terms of the median of each bj,ALL + bj,SPC. We demonstrate individual-tree performance for two species, Fraxinus lanuginosa and Magnolia kobus, as examples, because these two species show that species responses are similar to the common response and that the range of shoot dimensions between these species showed no large differences.


Most species had a positive response to tree height in the allometry 1 between total leaf-blade mass and non-leafy part mass (Fig. 2A). This means that leaf-blade mass supported by a given mass of non-leafy parts increased with tree height (Figs 3A and and4A).4A). Because the species response was positive in Sorbus alnifolia (Sa; Fig. 2A), S. alnifolia increased leaf-blade mass for a given non-leafy mass with tree height to a greater extent than other species (Fig. 4A). No effects of tree height were observed in the case of Acer amoenum (Aa) and Mongolia obovata (Mo) in the allometry between total leaf-blade mass and non-leafy part mass. However, their posterior distributions of b1,ALL + b1,SPC were largely biased toward the positive domain of the tree-height effect (Fig. 2A).

Fig. 2.
Response of shoot allometries to tree height in terms of 95 % credible intervals of posterior distribution in parameter bj of allometry j (j = 1 to 4). (A) Allometry 1 between total leaf-blade mass F and mass of non-leafy parts S; (C) allometry 2 between ...
Fig. 3.
Observed allometries between a pair of shoot dimensions in Flaxinus lanuginosa (circles) and Magnolia kobus (triangles). (A) Total leaf-blade mass F and non-leafy part mass S; (B) total leaf-blade area A and total leaf-blade mass F; (C) stem basal diameter ...
Fig. 4.
Relationships between tree height and predicted values for each shoot dimension. (A) Total leaf-blade mass, F, (B) total leaf-blade area, A, (C) stem basal diameter, D and (D) stem length, L, for the observed mean mass of non-leafy parts S (1·96 ...

Each species showed a negative response to tree height in the allometry 2 between total leaf-blade area and total leaf-blade mass (Fig. 2B). Therefore, leaf-blade area for a given leaf-blade mass, i.e. SLA, decreased with tree height (Figs 3B and and4B).4B). The species response of Ostrya japonica (Oj) was negative, and that of A. amoenum (Aa) was positive (Fig. 2B). This means that the degree of decrease in SLA with tree height was smaller in A. amoenum, and was larger in O. japonica than in the other species (Fig. 4B).

There was no tree-height dependence in the allometry 3 between stem basal diameter and total leaf-blade area, because most species responses were categorized into the no-effect group (Figs 2C, C,3C3C and and4C).4C). A. amoenum (Aa), M. obovata (Mo) and Tilia japonica (Tj) showed a positive response to tree height (Fig. 2C), which suggests that these species increased stem basal area for a given leaf area with tree height.

Most species showed a negative response to tree height in the allometry 4 between stem length and stem basal diameter (Fig. 2D). In other words, stem length for a given stem diameter of a leader shoot decreased with tree height (Figs 3D and and4D).4D). Because the species response was negative in M. obovata (Mo; Fig. 2D), M. obovata decreased stem length with tree height for a given stem diameter to a larger extent than the other species (Fig. 4D). Effects of tree height were not observed in Acer mono (Am) and T. japonica (Tj) in the allometry between stem length and stem basal diameter; however, their posterior distributions of b4,ALL + b4,SPC were largely biased toward the negative domain of the tree-height response (Fig. 2D).


This paper addresses the issue of how the functional morphology of current-year shoots changes with tree height under full-light conditions through observations of the leader-shoots at the tops of trees from stands of variable age after large-scale deforestation. Nabeshima and Hiura (2008) examined the tree-height dependence of fully exposed leaves of three Acer species in the same forest studied here. They reported that SLA decreased and area-based maximum photosynthetic rate remained unchanged with tree height. As leaves are developed on current-year stems, we examined how change at the level of a single leaf is associated with a whole current-year shoot. All of the 13 species studied demonstrated similar morphological responses to tree height at the scale of their current-year leader shoots, i.e. the leaf-blade mass supported by a given mass of non-leafy parts increased and SLA and stem length for a given stem diameter decreased with tree height, under full-light conditions. This is the first report to show that all co-occurring tree species show convergent performance of height-dependent plasticity at the scale of open-grown current-year shoots.

We expected, from our result in the allometry between stem basal diameter and total leaf-blade area, that the sapwood area that supports total leaf area at the scale of current-year shoots is proportional to the total leaf area irrespective of tree height. This is because we did not find a height dependence in the allometry between total leaf-blade area and stem basal diameter of current-year shoots (the power of allometry varied from 0·353 to 0·489, cf. Table S3) and because pith area does not change with tree height within species (Ambrose et al., 2009). Smaller powers than 0·5 can be attributed to the small size of current-year shoots that commonly have a higher percentage of pith in cross-sectional area of stem, which does not contribute to sap flow. Interspecific variation in the powers would depend on variation in pith size. At the present study site, Nabeshima and Hiura (2008) showed that leaf-specific hydraulic conductivity does not change with tree height across three Acer species, A. amoenum, A. japonicum and A. mono. This is possibly correlated with our expectation that basal sapwood area of a shoot relative to total leaf-blade area remains unchanged with tree height. At a mesic site such as TOEF where most rainfall occurs in summer, the pipe model relationship (Shinozaki et al., 1964) is a major constraint that connects a stem and its leaves because hydraulic conductivity is not largely limited by site conditions. This trend contrasts with the increase in stem sectional area relative to total leaf area with tree height under drier conditions at the scale of a small branch having no heartwood (Ambrose et al., 2009).

Recent studies have shown that SLA decreases with tree height (Nabeshima and Hiura, 2008; Woodruff et al., 2008; Ambrose et al., 2009; Mullin et al., 2009). Our result for all of the 13 species is consistent with these findings. Therefore, per unit of basal stem area, there is a larger load of leaf mass for a shoot at a higher position because the total leaf area per unit of basal stem area does not change with tree height. The decrease in stem length per unit of basal stem area is therefore necessary to retain sufficient mechanical strength to support a heavy leaf load (Cannell et al., 1988). From the view of water transport within a stem, the stubby shoot stems in tall trees contribute to a reduction in xylem length and conductive resistance (Normand et al., 2008). The shoot stem, especially the stem of a leader shoot, should provide a foundation for foliage expansion and crown development over several decades. The height-dependent change in the stem shape of a shoot suggests a decline in the increment rate of height. Asymptotic height growth in trees with large trunk diameters has been frequently observed (Poorter et al., 2006; Aiba and Nakashizuka, 2009).

In forest stands, air temperature, moisture and mechanical stresses change with tree height (Baldocchi et al., 2002). We chose open-grown trees that were not completely isolated from their neighbours, which allowed us to examine the effects of these ambient conditions on shoot morphology to gain a better understanding of shoot-level associations between leaf and stem. As wind speed is expected to be greater in taller tree crowns, there are also effects of wind exposure on shoots in terms of water and mechanical stresses (Martin et al., 1999; Baldocchi et al., 2002; Anten et al., 2010). Leaves developed under high water stress have small and tightly packed cells with thick cell walls. This results in small SLA and high mechanical toughness against wilting (Wright and Westoby, 2002; Poorter et al., 2009). This mechanism holds in the context of a height-related decrease in SLA within species (Thomas and Winner, 2002), because the leaves need to maintain photosynthetic capacity per unit of transpiring area in dry environments on tall crowns, as examined by Nabeshima and Hiura (2008). Changes with tree height in air temperature around leaves, which are partially caused by changes in exposure to wind, are also associated with leaf cell morphology (Poorter et al., 2009). Under mechanical stress, the stem decreased in length and increased in diameter to enhance its strength and stiffness (Pruyn et al., 2000). Wind exposure influences both stem length and the ratio of total leaf area to stem basal area at the scale of current-year shoots, which suggests that the morphological plasticity of leaf and that of stem are integrated responses to mechanical and hydraulic constraints (Garcia-Verdugo et al., 2009).

As a consequence of the shoot-level association between SLA and stem shape, the mass allocation to leaf blades in a shoot increased with tree height, and this association was common across the species, in accordance with our hypothesis. At the scale of the shoot, leaf-blade mass F, under mean non-leafy part mass S = 1·07 g, increased from 2·01 to 3·76 g in Fraxinus lanuginosa with increasing heights from 1 to 20 m. In contrast, at the scale of a tree or of a whole forest stand, the mass allocation to foliage leaves in the above-ground mass decreases with height. For example, the ratio decreased from 0·36 to 0·07 in Abies veitchii stands with increasing tree heights from 0·69 to 16·3 m (Tadaki et al., 1970). In a simulation study, the model that assumes a fixed mass allocation between leaves and a stem in a current-year shoot with tree height results in a failure of tall and exposed trees relative to short and shaded trees with respect to crown development (Kubo and Kohyama, 2005). This is because the increment of shoot populations in tall trees cannot maintain sufficient amounts of leaf to meet the increasing demand for construction cost of non-leafy parts. The height-dependent increase in mass allocation to leaves at the scale of a current-year shoot compensates for the increasing construction costs at the scale of a whole tree. Leaf- and shoot-level plasticity are regulated by the whole-tree-level carbon economy for survival and height increment, which suggests a shoot-level constraint and tree-level requirement mutually contribute to their dynamic and functional performances. Species that do not show height dependence in mass allocation in their leader shoots possibly increase mass allocation to leaves with increasing tree height at the scale of a whole population of current-year shoots. One probable way to achieve regulation at the scale of the shoot population is for the ratio of the numbers of short shoots to long shoots to increase with tree height (Maillette, 1982; Wilson, 1991). This height-dependent change contributes to the maintenance of the amount of leaves in a whole tree because the mass ratio of leaf blades to stem is greater in short shoots than in long shoots (Yagi and Kikuzawa, 1999; Yagi, 2004).

The overall morphological dependence of open-grown current-year shoots on tree height and shoot position is generalized as follows. Shoots in tall trees have a larger mass proportion of leaf blades, lower leaf area per leaf mass and stubbier stems than in short trees. These properties were shared among eight co-occurring canopy species and five sub-canopy species in terms of maximum tree height. We conclude that the association between leaf and stem underlies mechanical and physiological functions of current-year shoots over a range of tree heights, and during the course of tree growth. The present results give new insight into the functional constraints associated with tree-height development in which the height dependence of shoot morphology contributes to successful crown expansion and height gain.


We thank K. Hikosaka, M. Suzuki, T. Hiura, I. Terashima, T. Saito, K. Makoto and N. Eguchi for their invaluable comments. Constructive comments were provided by B. Shipley and two anonymous reviewers. We are grateful to T. Ishii and the technical staff of TOEF for their support in the field, S. Tanabe, T. Shirasaki and the members of TOEF for their help and suggestions during this work. This work was partially supported by grants from the Ministry of Education, Science, Sports and Culture of Japan (nos. 19405006, 21405006).


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