Effects of turnover and internal variability of treeroot systems on modelling coarse rootarchitecture: comparing simulations for youngPopulus deltoides with field data

Pekka Nygren, Miaoer Lu, and Harry Ozier-Lafontaine

Abstract: The behaviour of the FracRoot model, which describes coarse-root architecture based on proximal root diame-ters using a recursive algorithm, was studied using field data on Populus deltoides Bartr. ex Marsh. in a short-rotationplantation. Roots were described as a branching, hierarchical network of connected links. Diameters of daughter linkswere estimated from the diameter of the mother link using scaling parameter p and allocation parameter q, which werebased on sharing of the root cross-sectional area by daughter links. Parameters were estimated from complete root excava-tions. The length of each link and the vertical and horizontal branching angle distributions were recorded. Parameter p val-ues were distributed log-normally, and q values followed a beta distribution both for the whole root system and within 5mm link diameter classes. Including the variability of p and q in the model did not significantly improve root length esti-mates compared with the use of mean p and q values over all branching points. Including a coarse-root turnover factorbased on field-observed evidence on root mortality improved the model fit to field data. Root length was more sensitive toparameter values and turnover factor than root mass. Field observations and the importance of the turnover factor to simu-lation accuracy suggest that coarse-root turnover should be considered in root research, at least under conditions of strongcompetition or other external stress.

Resume : Le comportement du modele « FracRoot » qui simule l’architecture des racines de transport des arbres a partirdu diametre des racines du collet a l’aide d’un algorithme recursif a ete etudie avec des donnees de terrain sur le Populusdeltoides Bartr. ex Marsh. cultive en rotations courtes. Les racines ont ete decrites sous la forme d’un reseau hierarchise etramifie de liens connectes. Le diametre des liens fils a ete estime a partir du diametre du lien mere en utilisant un para-metre d’echelle p et un parametre d’allocation q bases sur le partage de la surface radiale entre les liens fils. Les para-metres ont ete estimes apres une excavation complete du systeme racinaire. La longueur de chaque lien ainsi que ladistribution des angles verticaux et horizontaux des ramifications ont ete mesurees. Le parametre p suivait une distributionlog-normale et le parametre q une distribution de type Beta dans l’ensemble du systeme racinaire jusqu’aux classes de dia-metres des liens de 5 mm. Le fait de tenir compte de la variabilite de p et q n’a pas significativement ameliore l’esti-mation de la longueur des racines comparativement aux simulations effectuees en appliquant la valeur moyenne de p et qa tous les points de ramification. Par contre, l’introduction d’un facteur qui tient compte du renouvellement des grosses ra-cines sur la base d’indices de mortalite des racines observes sur le terrain ameliore nettement la precision du modele. Lalongueur des racines est plus sensible a la valeur des parametres p et q et au facteur de renouvellement que la biomassedes racines. Les observations realisees sur le terrain et l’importance du facteur de renouvellement des grosses racines pourla precision du modele indiquent qu’on devrait tenir compte du renouvellement des grosses racines dans les recherches surles racines, tout au moins dans des conditions de forte competition ou d’exposition a des stress externes.

[Traduit par la Redaction]

Introduction

Access to soil resources, water and nutrients, by tree rootsis defined by both root architecture and activity. Coarse-rootarchitecture can provide an estimate of the influence area of atree (i.e., the radial extension and depth of soil available forwater and nutrient acquisition) because nutrient-absorbingfine roots attached to coarse transport roots are generallyquite short and external mycelia of arbuscular mycorrhizae,with which most tree species form symbioses, do not growfar from the host root. Direct measurement of root architec-ture is too laborious for most practical applications for whichthis information would be valuable, like determining optimaltree spacing in managed forests, plantations, or agroforestrysystems. Thus, developing tools for describing root systems

Received 27 May 2008. Accepted 6 October 2008. Published onthe NRC Research Press Web site at cjfr.nrc.ca on 19 December2008.

P. Nygren.1 Department of Forest Ecology, P.O. Box 27, 00014University of Helsinki, Finland.M. Lu.2 Department of Forestry, 203 ABNR Building,University of Missouri, Columbia, MO 65211, USA.H. Ozier-Lafontaine. Unite Agropedoclimatique de la ZoneCaraıbe, INRA Antilles-Guyane, Domaine Duclos (Prise d’eau),97170 Petit-Bourg, Guadeloupe, French Antilles.

1Corresponding author (e-mail: pekka.nygren@helsinki.fi).2Present address: 1075 W. State Street, G102, Redlands, CA92373, USA.

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Can. J. For. Res. 39: 97–108 (2009) doi:10.1139/X08-158 Published by NRC Research Press

based on a few relatively easily measured parameters is ofgreat interest.

Halle and Oldeman (1970) were the first to show that thevast diversity of tree canopy forms may be reduced to a fewarchitectural models. More importantly, they showed that alltrees are composed of a few repeated basic structures. Thisidea has significantly advanced the development of morpho-logical (Room et al. 1994; Godin et al. 1999) andfunctional–structural tree models (Sievanen et al. 2000).Root architecture is also conceivably formed by repeated el-ementary units (Sievanen et al. 2000). Fractal geometry,which was developed for describing the natural forms thatcannot be accommodated by Euclidean geometry (Mandel-brot 1983), has proven to be an important mathematical toolfor describing tree root architecture (van Noordwijk et al.1994; Berntson 1996).

Van Noordwijk et al. (1994) proposed that a fractal rootarchitecture model could be parameterized according to thepipe model theory (Shinozaki et al. 1964). This approachhas been tested for roots and shoots of several tropical treespecies (van Noordwijk and Purnomosidhi 1995; van Noord-wijk and Mulia 2002). Building on the numerical version ofthe model (Spek and van Noordwijk 1994), Ozier-Lafontaineet al. (1999) developed a recursive algorithm, the FracRootmodel, for reproducing and visualizing complete-tree rootsystems based on proximal root characteristics.

Comparison of FracRoot results with field data indicatedthat the model tended to overestimate root mass (Ozier-Lafontaine et al. 1999). Two possible causes for this ten-dency are that (1) the model does not take into accountroot turnover but simulates an idealized root system and(2) the model is based on the self-similarity assumption,i.e., that the root system maintains a similar form over arange of scales. Coarse roots of trees are assumed to bequite stable organs with a low turnover rate. However,coarse-root senescence over years may arise because ofadverse growing seasons, herbivory, pathogens, and shootdamage like defoliation (Salas et al. 2004; King et al.2007). Further, if nutrient uptake of a tree seriously de-pletes nutrients from parts of the soil, fine roots die (vanNoordwijk et al. 1996) and coarse transport roots, towhich the dying fine roots are attached, may also die.This process is analogous to self-pruning of branches in atree canopy when they become too heavily shaded.

The self-similarity assumption may be criticized on amathematical basis (Mandelbrot 1983) and based on the ob-servation that link radii and length tend to scale by differentfactors (West et al. 1999; Salas et al. 2004), thereby leadingto nonsimilar variation in link volume. Most natural formsare statistically self-similar over a limited range (Avnir etal. 1998). In transport networks like coarse roots, structuralcomplexity translates into asymmetric branching patterns,and multifractal analysis of root structures has been pro-posed as a suitable tool for providing insights on the abilityof root systems to forage in time and space for soil resour-ces. Fractal dimensions of a multifractal system vary de-pending on the processes that regulate their formation(Berntson 1996). Parameter values of FracRoot-type modelshave shown large overall variation, although no trends re-lated to root topology or diameter have been observed (vanNoordwijk and Purnomosidhi 1995; Ozier-Lafontaine et al.

1999; Salas et al. 2004). However, the use of mean parame-ter values may cause bias in the simulations of root-systemcomponents. The first way to approach the multifractal na-ture of tree root systems may be based on the analysis oftheir internal variability and may require that the mean pa-rameter values be replaced with distribution laws in themodel.

The aim of this contribution is to test the effects ofturnover and internal variability of tree root systems onmodelling coarse-root architecture. The test data are basedon measurements on root systems of eastern cottonwood(Populus deltoides Bartr. ex Marsh.) growing in a temper-ate short-rotation plantation. The specific objectives of thestudy were to (1) test whether the accuracy of the rootarchitecture model FracRoot (Ozier-Lafontaine et al.1999; Salas et al. 2004) can be improved by introducinga stochastic root turnover factor to the model; (2) testwhether replacing mean parameter values with distribu-tions laws improves the accuracy of the model for rootsimulation; and (3) evaluate the combined effects of turn-over and parameter distributions on the accuracy of themodel.

The FracRoot modelHere, the concept of link will refer to a piece of a root

between two branching points, or an internode. An axis isformed by subsequent links; especially, the main axis of aroot is formed by the thickest link attached to a proximalroot and is always continued by the thickest link in abranching point. A proximal root is a link directly attachedto the root collar. The concept of root (sometimes individualroot for clarity) will refer to the main axis starting from aproximal root and all laterals attached to it. Root systemwill refer to all roots of a tree.

A tree root is described as a continuously branching hier-archical network of connected links. Scaling factors betweensubsequent links were assumed to be invariant within awhole tree in the original FracRoot model formulation(Spek and van Noordwijk 1994; van Noordwijk et al. 1994;Ozier-Lafontaine et al. 1999), or the root system was as-sumed to be a self-similar fractal. The branching generationof a link (West et al. 1997) is assigned following a develop-mental sequence (cf. Berntson 1996), i.e., starting from theroot collar with stem as generation 0 and proximal rootshaving branching generation 1.

Root architecture is designed using a recursive algorithm(Ozier-Lafontaine et al. 1999). The algorithm is applied untilthe final branch of the network, links of minimum diameterDm, are reached. Here, we applied Dm = 2 mm, which wasassumed to be the limit between coarse and fine roots(Block et al. 2006). The scaling factor p is defined as theratio of the square of link diameter before bifurcation (D2

b)to the sum of squares of the diameters of bifurcating daugh-ter links (

PD2

a):

½1� p ¼ D2bPD2

a

The allocation factor of a link’s cross-sectional area, q, isestimated as

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½2� q ¼ maxðD2aÞP

D2a

Except for the largest link, all other daughter links havethe same diameter. The diameter of the largest daughterlink (Damax) is estimated as

½3� Damax ¼qDb

p

and the diameters of all other daughter links are estimatedas

½4� Da ¼Damaxð1� qÞqðn� 1Þ

where n is the total number of daughter links formed in thebranching event. Each new link has a random bifurcationangle generated from the cumulative bifurcation angle fre-quency (Salas et al. 2004). Link length is estimated as alogarithmic function of its own branching generation. Newlinks with a diameter greater than Dm have further bifurca-tions generated through the process described above.

The number of new links is estimated from the percen-tages of two-, three-, and four-daughter links in the fielddata. FracRoot simulations without a turnover factor arecomputed so that these percentages total 100%. A root turn-over factor accounting for the occurrence of ‘‘zero links’’ isincluded in the model; this occurrence refers to an abnormaltermination of a root before it reaches Dm. In simulations in-cluding root turnover, the sum of the percentages of zero,two, three, and four links formed in a branching event total100%. The number of daughter links is randomly selectedfrom the cumulative frequency range of the number of linksformed in a branching event.

In all earlier applications of FracRoot-type models, invar-iant p and q values were used in the whole root system (vanNoordwijk and Purnomosidhi 1995; Ozier-Lafontaine et al.1999; Salas et al. 2004). In this work, six alternative valuesfor parameters p and q were tested:

(1) general mean of the measured parameter values for thewhole root system;

(2) general median of the measured parameter values for thewhole root system;

(3) estimation of parameter values from general distributionsfor whole root system, based on observed distributionsof p and q in the field;

(4) mean of the measured parameter values by 5 mm dia-meter classes of field data;

(5) median of measured parameter values by 5 mm diameterclasses of field data; and

(6) estimation of parameter values from observed distribu-tions by 5 mm diameter classes of field data.

The values of p and q are used in eqs. 1–4 for estimatingthe diameters of daughter links. In cases 3 and 6, the p andq values are randomly sorted in each branching event fromthe respective distribution corresponding to the mother linkdiameter.

This version of FracRoot software was developed withC++ programming language. OpenGL graphic libraries wereused for visualization of root architecture.

Field dataThe parameterization and validation data for P. deltoides

were measured in an experimental plantation on the Mis-souri River floodplain at the University of Missouri’s Horti-culture and Agroforestry Research Center in New Franklin,Missouri, USA (39801’N, 92846’W; 197 m altitude). Thegrowing season is from April to October. Annual precipita-tion is ca. 1130 mm, with a slight peak in spring and earlysummer. Water deficits develop in late summer. The fieldfloods occasionally in spring and early summer. The soil isa Moniteau silt loam (fine-silty, mixed, superactive, mesicTypic Endoaqualf), which is fertile, moderately well-drained, and permeable (Pallardy et al. 2003). The plantationwas established using 20 cm long cuttings at 1 m � 1 mspacing in May 1999 (Pallardy et al. 2003). Three P. del-toides clones were sampled. The mean height of P. deltoidestrees was 8.4 m in autumn 2002.

The root systems of two trees of each P. deltoides clonewere studied in autumn 2002. All proximal roots of a treewere exposed to the first branching point. Link diameter atthe root collar and before the first branching point, azimuth,and inclination of the proximal root were recorded.

One or two roots were selected from each tree for com-plete excavation. The roots were selected so that when thecompletely excavated roots of all trees were pooled, theycovered the observed diameter range of proximal roots ofthe sample trees. The roots were carefully unearthed one byone using small hand tools to avoid changes in their posi-tion. The excavation proceeded along the main axis. All lat-erals attached to the main axis were also followed andexposed until a minimum diameter of 2 mm. All axes werefollowed as far and deep as they had grown. The data meas-ured on these roots included:

(1) branching generation according to West et al. (1997);(2) link diameter before each bifurcation, and diameters of

the bifurcating links;(3) link length, Li;(4) vertical and horizontal angle between the bifurcating

links (bifurcation angle); and(5) relationship between fresh root volume and dry mass

(basic density) from a sample of 45 root pieces of differ-ent diameters.

After the excavation of individual roots, all soil and rootswere removed to a depth of 1 m in a 1 m2 square centred onthe stem around five P. deltoides trees. This allowed us toobtain an adequate estimate of tree root biomass, with rootsof nonsample trees growing into the excavated volume com-pensating for roots of sample trees growing outside the 1 m3

zone (Scarascia-Mugnozza et al. 1997). Roots growing be-low 1 m depth were limited to a few brace roots that termi-nated less than 25 cm beyond the 1 m excavation limit.Roots were separated from soil by hand, washed free of soilusing a 1 mm sieve, and oven-dried at 70 8C until reachinga constant mass.

Data analysesThe turnover factor was determined by counting the rela-

tive frequency of the cases when a root terminated beforereaching the minimum diameter of 2 mm. The terminationof a root before 2 mm was assumed to indicate turnover be-

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cause normally the roots continued well past this diameterand formed abundantly bifurcating fine roots. The turnoverfactor as defined here refers to a missing link in a casewhere one is expected and integrates link turnover over thelife-span of a tree.

The fit of several probability distributions to observed dis-tributions of parameter p and q values in different link diam-eter classes was tested using a one-sample Kolmogorov–Smirnov test. Distributions that fit best to all link diameterclasses were selected for further model test runs.

Total length of each fully excavated root was computedfrom the field data by summing up the length of all links ofthe root. The model was tested in the six cases described in‘‘The FracRoot model’’ section, running them with and with-out root turnover. Because of the stochasticity of the model,the mean of 75 runs of the software was used as the simu-lated value in all cases. The accuracy of root length predic-tion was evaluated by fitting a linear regression withoutintercept between observed and simulated root lengths. Asignificant regression slope close to unity (0.95–1.05) wasthe main criterion for the model performance in this case(Mayer and Butler 1993; Rita and Ekholm 2007), with thecoefficient of explained variance (R2) as an auxiliary metric.The same procedure was used for estimating the accuracy ofthe prediction of the number of links in a root.

Root mass was simulated by computing the volume ofeach link from its radius and length, multiplying the volumeby basic density for link mass, and summing up link massesfor whole root system mass. Because we did not determinethe mass of individual roots, we plotted the simulated rootsystem mass against the measured root system mass and per-formed the same analyses used for individual root length.

Results

Parameter estimatesThe mean and median values of the scaling parameter p

and allocation parameter q did not coincide either for thewhole root system or for the 5 mm wide link diameterclasses (Table 1). This result suggests that the distributionsof the parameters are not normal. Statistically significantdifferences in p values among diameter classes were not de-tected by Kruskall–Wallis (P = 0.1263), Brown–Mood Me-dian (P = 0.0578), or Savage (P = 0.4362) nonparametrictests. All three tests indicated statistically significant differ-ences among diameter classes for q values (P = 0.0029 orlower). Because of the distribution form, a Kruskall–Wallistest is more reliable for p than q, a Savage test is more reli-

able for q, and the Brown–Mood Median test approximatelysuits both. No trend in parameter values according to linkdiameter was evident, e.g., the two highest median p valueswere observed in diameter classes >2.5 and 1.0–1.5 cm, andthe lowest and second lowest mean q values were measuredin the smallest and largest diameter class, respectively.

The p values by diameter class followed a log-normal dis-tribution with a threshold parameter q (Fig. 1), and q valuesfollowed a beta distribution that was restricted between 0.25and 0.99 (Fig. 2). The lower limit for the beta distribution isthe theoretical minimum value of q when four new links, themaximum number in our simulations, are formed at abranching point (eq. 2). The upper limit was determinedfrom field data. According to the one-sample Kolmogorov–Smirnov test, the fit of the log-normal distribution suc-ceeded for p in all diameter classes (P < 0.05). The fit ofthe beta distribution for q failed only for diameter class0.5–1.0 cm.

We also tested whether the form of distributions fitted bydiameter class differed among each other by comparing thecumulative distributions by diameter class using a two-sampleKolmogorov–Smirnov test (P < 0.05). Statistically significantdifferences were detected between several diameter classesfor p (Table 2). In the case of q, the only significant differencewas between diameter classes 1.0–1.5 and >2.5 cm, while theq values of the other classes were intermediate (Table 3).

Although link length varied considerably within eachbranching generation (Fig. 3), we observed a statisticallysignificant trend of link length as a function of the natu-ral logarithm of branching generation. This regression wasused for estimating link length in all FracRoot simula-tions.

The distribution of the vertical bifurcation angles relativeto the mother link had a strong peak around 08 (Fig. 4). Oth-

Table 1. Mean and median of scaling parameter p and alloca-tion parameter q by link diameter class in Populus deltoides.

Mean Median

Diameter class (cm) n p q p qD £ 0.5 47 1.1689 0.6998 1.0889 0.69230.5 < D £ 1.0 51 1.1642 0.8414 1.0622 0.90001.0 < D £ 1.5 22 1.1528 0.8725 1.1301 0.92471.5 < D £ 2.0 21 1.0530 0.8073 1.0471 0.89522.0 < D £ 2.5 12 1.0911 0.8212 1.0615 0.8435D > 2.5 27 1.1964 0.7777 1.1890 0.8729General 180 1.1510 0.7933 1.0900 0.8621

Fig. 1. Log-normal distribution density functions of scaling para-meter p by link diameter class in Populus deltoides. The distribu-tions are drawn using parameter values given in Table 2.

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erwise the distribution was quite flat with low frequenciesfor large up- or downward angles. The distribution of thehorizontal bifurcation angles of both species peaked at 08and was otherwise flat (Fig. 5).

According to the field estimations, 31% of links predictedby the ideal root structure were missing in P. deltoides.These missing links were interpreted as root turnover. Two,three, and four or more new links were observed in 42%,16%, and 11% of branching points, respectively. These per-centages sum to 100% with the turnover percentage. The ba-sic density of P. deltoides roots was 0.38 g�cm–3.

Model test runsThe means and medians of p and q given in Table 1 and

the distributions defined by parameters given in Tables 2and 3 were used in FracRoot simulations. The FracRootmodel produced high overestimations of root length of P.deltoides when the root turnover factor was not considered(Fig. 6). Overestimation of root system mass was much

more moderate, ranging from an overestimation of 1.13times for case 1 to 1.78 times for case 3. When root turn-over was considered, the regression slope between observedand estimated root length differed significantly from 1 onlyfor case 5, mainly because of relatively high standard devia-tions of the slopes (Table 4). However, the slope differed by5% or less from unity only in cases 1 and 3. These resultsindicate that root turnover affected the reliability of rootlength estimates provided by the FracRoot model more sothan did the stochastic properties of parameters p and q.

Because of the strong root length overestimation whenroot turnover was not taken into account, only estimateswith turnover were evaluated for the number of links(Table 5) and root system mass (Fig. 7). Parameter valuesdefined for the whole root system (cases 1–3) produced

Fig. 2. Beta distribution density functions of allocation parameter qby link diameter class in Populus deltoides. The distributions aredrawn using parameter values in Table 3.

Table 2. Parameters of log-normal distribution for scalingparameter p by link diameter class in Populus deltoides.

Diameterclass (cm) q m s

Significanceletter

D £ 0.5 0.2912 –0.2645 0.5290 a0.5 < D £ 1.0 0.6850 –0.9321 0.6352 a1.0 < D £ 1.5 0.6459 –0.7438 0.3761 b1.5 < D £ 2.0 0.5248 –0.6706 0.2646 c2.0 < D £ 2.5 0.7693 –1.2547 0.5372 bD > 2.5 0.8962 –1.3284 0.5082 bGeneral 0.1885 –0.0853 0.3036

Note: Distributions for diameter classes followed by the sameletter do not differ significantly (P £ 0.05) according to the two-sample Kolmogorov–Smirnov test.

Table 3. Parameters of beta distribution for alloca-tion parameter q by link diameter class in Populusdeltoides.

Diameter class(cm) a b

Significanceletters

D £ 0.5 2.4289 1.5228 ab0.5 < D £ 1.0* 3.4750 0.8858 ab1.0 < D £ 1.5 3.8552 0.7459 a1.5 < D £ 2.0 1.8975 0.6255 ab2.0 < D £ 2.5 2.4619 0.7552 abD > 2.5 2.683 0.710 bGeneral 2.1147 0.7655

Note: All distributions are restricted between 0.25 and0.99. Distributions for diameter classes followed by thesame letter do not differ significantly (P £ 0.05) accordingto the two-sample Kolmogorov–Smirnov test.

*The observed distribution differs significantly from atheoretical beta distribution according to the one-sampleKolmogorov–Smirnov test.

Fig. 3. Mean link length (L) as a function of branching generation(i) in Populus deltoides. The error bars indicate the 95% confidenceinterval of the mean. The trend line fitted to data is L = 28.424 –5.0423 � ln(i) (P = 0.0040; r2 = 0.48).

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more accurate estimates for the number of links within aroot than did diameter-class-specific parameter values (cases4–6; Table 5). However, the number of links was underesti-mated by 14% (case 1) or more in all cases. The mean of pand q for whole root system produced the best estimates forroot length (Fig. 6, Table 4) and number of links in a root(Table 5) but the poorest estimate for root system mass(Fig. 7).

In spite of being highly significant, all regressions hadlow R2 values. This indicated that the FracRoot model hada limited power to predict individual-root characteristics butthe significant slope close to unity suggested that total rootsystem length of a tree could be estimated quite accurately,as over- and underestimations for individual roots offseteach other at the tree level. The error in the total number oflinks within a root system fell between those of length andmass estimates.

Because the FracRoot model appeared to be more sensi-tive to the turnover factor than to internal variability of pa-rameters p and q, a sensitivity analysis was conducted forthe case that gave the best root length and link number esti-mates of P. deltoides, i.e, the use of mean p and q values forthe whole root system with turnover. The turnover factorwas varied by 3 and 6 percentage units around the observedvalues; these variations represented about 10% and 20%changes relative to the observed turnover factor. Root lengthand number of links appeared to be more sensitive to theroot turnover factor than did root system mass (Table 6). Aturnover factor of 28% (a 3 percentage unit decrease) pro-duced a better estimate for the number of links (slope differ-ing by less than 5% from unity) than did the observed value,i.e, 31%. The former case resulted in a 5% overestimation inroot length, as opposed to a 4% underestimation, andslightly improved the root system mass estimate (Table 6).When all three variables are considered together, the 28%turnover factor appeared to produce better estimates for root

system characteristics than did the factor estimated fromfield data, 31%.

Prediction of root system characteristicsThe mean simulated root system length of five P. del-

toides trees varied from 74 to 95 m per tree (Table 7). Themedian length of the whole root system estimated by thesimulations was 80 m (tree WS). Several small proximalroots produced almost the same total root system length asfewer large proximals; 20 proximals with a mean diametersquared (D2

pr) of 1.77 cm2 in the WS tree and 14 proximalswith a mean D2

pr of 4.33 cm2 in the WM tree resulted in to-tal root system lengths of 80 and 85 m, respectively. Fur-ther, the confidence intervals of the root system lengths ofthese trees overlapped (Table 7), indicating a nonsignificantdifference (Rita and Ekholm 2007). The 95% confidence in-tervals were 7.0%–8.4% of mean root system length, withthe maximum of 9.5% for tree WM, and 5.1%–5.9% ofmean root system mass, with the minimum of 2.6% for treeWS (Table 7).

Visualization of P. deltoides root system was computedfor tree YM. Because FracRoot generates the root systemstochastically, visualization of a case with root systemlength close to the mean of the 75 model test runs for thesame tree was selected here (Fig. 8). The root system of P.deltoides was restricted to quite a small area around thestem. The simulated influence area of the tree was ellipsoi-dal (Fig. 8a) with a long axis of 3.0 m and a short axis of1.7 m. The maximum depth of the coarse-root system wasabout 0.9 m. However, the top 20 cm of soil appeared tohave a high coarse-root density. Note that the visualizationtends to exaggerate the number of deep roots, becausesuperficial roots are superposed in the graph, while eachdeep root is shown individually (Fig. 8b).

Fig. 4. Distribution of vertical bifurcation angles of daughter linksrelative to the mother link in Populus deltoides. Positive values in-dicate downward angles relative to the mother link and negativevalues indicate upward angles.

Fig. 5. Distribution of horizontal bifurcation angles of daughterlinks relative to the direction of the mother link in Populus del-toides. Negative angles are to the left relative to mother link andpositive angles are to the right.

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Discussion

Coarse-root turnoverWhile a high fine-root turnover rate is a generally ac-

cepted feature of root systems (Hendricks et al. 1993; Blocket al. 2006), coarse-root turnover has received much less at-tention. Coarse-root turnover is often considered to becaused by external stress factors such as diseases and fire(King et al. 2007). Data on coarse-root turnover rate or nec-romass are seldom reported in literature. In Pinus kesiyaRoyle ex Gordon plantations, coarse-root necromass ac-counted for about 10% of biomass in 15- and 23-year-oldplantations and about 20% in a 6-year-old plantation (Johnet al. 2002). Strong competition in the dense P. deltoidesstand and in the youngest P. kesiya plantation (John et al.2002) may have caused coarse-root turnover.

The FracRoot model reproduces a mathematically ideal-ized coarse-root system based on the proximal root data at

the moment of measuring the proximal roots. For example,if root turnover is not taken into account when determiningthe branching parameters from complete or partial excava-tion of the coarse-root system, FracRoot reproduces the rootsystem without any missing links, i.e., each link has two tofour daughter links as estimated from the observed frequen-cies. Thus a deterministic ‘‘perfect’’ root architecture isformed. Further, the recursive algorithm of the programmestops at a predetermined minimum diameter, but in the fielda sizable proportion of roots did not reach the 2 mm limitapplied in this study. Thus, we described the turnover factoras a case of ‘‘0 daughters’’ at the end of a mother link with adiameter greater than 2 mm. This method keeps the modelsimple compared with estimating root mortality based onroot length, topology, and age (Jourdan and Rey 1997).

Introducing the turnover factor based on the proportion ofassumed missing links in the field data considerably im-proved the agreement between the observed and simulated

Fig. 6. The relationship between observed and simulated root length in Populus deltoides when simulation is done using p and q values forthe whole root system (cases 1–3) without turnover (a), using diameter-class-specific p and q values (cases 4–6) without turnover (b), usingp and q values for the whole root system (cases 1–3) with turnover (c), and using diameter-class-specific p and q values (cases 4–6) withturnover (d). See ‘‘The FracRoot model’’ section for a description of the cases. The simulated values are means of 75 runs of the FracRootmodel. The slopes of the regressions are given in Table 4. ‘‘Dist’’ refers to the use of distribution laws for defining parameter values incases 3 and 6.

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root length in P. deltoides (Fig. 6). The FracRoot model wasalso quite sensitive to the turnover factor (Tables 4 and 6).The coarse-root turnover factor seemed to be the most im-portant single factor for improving the accuracy of rootlength estimates. The sensitivity of the model to the turnoverfactor also suggests that the turnover factor may be cali-brated for different cases by using Monte Carlo type itera-tion when it has not been estimated from field data. In fact,a combination of the estimates for root length, number oflinks in a root, and root system mass in case 1 suggestedthat the field-estimated turnover factor might have beenslightly overestimated, as reducing the turnover factor from31% to 28% improved the accuracy of the estimate for thenumber of links, while the estimate of root length changedfrom a small underestimation to a small overestimation(Table 6).

Internal variability of root system propertiesInclusion of the internal variability of scaling parameter p

and allocation parameter q without a root turnover factor re-sulted in root length overestimation by a factor of ca. 6.5 inP. deltoides (Fig. 6). Root mass was overestimated by amuch smaller factor. Ozier-Lafontaine et al. (1999) observedthat root system mass was overestimated by a factor of 1.3when using invariant mean p and q values without a turn-over factor for Gliricidia sepium (Jacq.) Kunth ex. Walp.

The current tests of the FracRoot model contained one ortwo stochastic factors: turnover, parameter distributions, orboth. As is typical for a stochastic model, FracRoot poorlydescribed characteristics of individual roots but gave a rela-tively good fit for the whole root system. This was espe-cially evident by the poor R2 values for regressionsbetween observed and simulated values for root length orthe number of links within a root (Fig. 6, Tables 4 and 5),indicating that these characteristics of an individual rootmay have been seriously under- or overestimated. Althoughthe slope of the regression between observed and estimatedvalues differed significantly from unity (Student’s t test)only in few cases, this result was mostly caused by quitehigh standard errors (Table 4). To avoid accepting largedifferences between data sets with high standard error asnonsignificant, a preset ‘‘biologically significant’’ differenceshould be used as the model selection criterion (Rita andEkholm 2007). Here, we decided to observe differencesof ±5% or less around unity as an indicator of acceptablemodel fit. This result also clearly demonstrated that Frac-Root or similar models should never be used for estimatingthe length of a root but rather the total length of a root sys-tem.

The parameter set (p, q, and turnover factor) and linklength equation (Fig. 3) that produced the best fit for rootlength underestimated root system mass of P. deltoides(Fig. 7). Including the stochasticity of p and q resulted in asmall improvement. Root basic density was quite invariantamong roots of different diameter. Thus, variation in basic

Table 4. The fit between the simulated and observed root lengths of Po-pulus deltoides according to the regression without intercept between theobserved and simulated link number.

Without turnover With turnover

Estimation ofparameter values Slope R2 Slope* SE R2

Same estimation criterion for whole root system(1) Mean 5.5913 0.71 0.9572 ns 0.0758 0.84(2) Median 7.9375 0.69 0.8266 ns 0.0944 0.65(3) Distribution 9.044 0.70 0.9492 ns 0.1161 0.61

Parameter estimation by link diameter class(4) Mean 4.8189 0.69 0.8556 ns 0.0763 0.78(5) Median 5.0602 0.74 0.7108 (0.05) 0.1012 0.46(6) Distribution 7.9236 0.70 0.8487 ns 0.1054 0.50

Note: Results for cases with six different estimates for parameters p and q withand without turnover are presented. See ‘‘The FracRoot model’’ section for a de-scription of the cases. The simulated values are means of 75 runs of the FracRootmodel. All regressions are significant at P = 0.0002 or lower. SE is the standarderror of the slope.

*The value in parentheses is the P value for H0: slope = 1 in two-tailed Student’s ttest. ns, nonsignificant. Slopes for the cases without turnover were not tested.

Table 5. The fit between simulated and observed number of linksin roots of Populus deltoides according to the regression withoutintercept between the observed and simulated link number.

Estimation ofparameter values Slope*

SE ofslope P R2

Same estimation criterion for whole root system(1) Mean 0.8602 ns 0.1091 <0.0001 0.63(2) Median 0.7317 (0.1) 0.1358 0.0010 0.20(3) Distribution 0.8005 ns 0.1511 0.0011 0.24

Parameter estimation by link diameter class(4) Mean 0.7181 (0.05) 0.0983 0.0002 0.54(5) Median 0.5858 (0.02) 0.1194 0.0017 0.16(6) Distribution 0.7266 (0.1) 0.1183 0.0005 0.29

Note: Results for cases with six different estimates for parameters pand q are presented. See ‘‘The FracRoot model’’ section for a descriptionof the cases. The simulated values are means of 75 runs of the FracRootmodel.

*The values in parentheses are P values for H0: slope = 1 in two-tailedStudent’s t test. ns, nonsignificant.

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density within the root system was an unlikely reason forthis model behaviour.

Root system mass also responded much more conserva-tively to changes in parameter values than did root length.There may be both mathematical and biological causes forthis model behaviour. First, root length and root systemmass were estimated at different scales, i.e., an individualroot for length and the whole root system for mass. Aspointed out earlier, length estimates for individual roots var-ied considerably, while the estimated mean for several rootsappeared to be more accurate, with under- and overestima-tions offsetting each other. Although we base our discussionon the mean of several roots, this value may still have morestatistical variation than the estimate for the whole root sys-tem mass. When whole root system length and mass weresimulated (Table 7), the greater variation in length estimatespersisted, as indicated by wider 95% confidence intervals.

Second, when considering the structure and functioning oftrees, this feature may have been caused by the fact that the

proximal roots and second and third branching generationsaccounted for most of root system mass. Because the bio-mass of proximal roots was estimated from their measureddimensions in FracRoot, up to 40% of root system massmay have been based on direct measurements, while an al-most negligible proportion of root system length was basedon measurements. Thus, increasing the number of roots ofhigh branching generations strongly affected the root lengthestimate but these small-diameter roots had only a slight ef-fect on root system mass. We considered the root systemlength to be an ecologically more meaningful variable thanmass (cf. Lynch 1995) and aimed for greater accuracy inroot length estimates.

Possibilities for improving the FracRoot modelEstimating link length is recognized as a difficult task in

FracRoot-type models (Salas et al. 2004). This was also evi-dent in the current data set, in which the case that producedthe best fit for root length (case 1: mean p and q for whole

Fig. 7. The relationship between observed and simulated root system mass in Populus deltoides when simulation is done using p and qvalues for whole root system (cases 1–3) with turnover (a) and using diameter-class-specific p and q values (cases 4–6) with turnover (b).See ‘‘The FracRoot model’’ section for a description of the cases. The simulated values are means of 75 runs of the FracRoot model. Allregressions are significant at P = 0.0008 or lower. The number in parentheses after the regression equation indicates the significance levelfor H0: slope = 1 in two-tailed Student’s t test. ‘‘Dist’’ refers to the use of distribution laws for defining parameter values in cases 3 and 6.

Table 6. Sensitivity of the estimates of root length, number of links within a root, and whole root system mass of Populus del-toides simulated with FracRoot model to root turnover factor according to the regression without intercept between the measuredand simulated values.

Root length No. of links Root system mass

Turnover factor Slope SE R2 Slope SE R2 Slope SE R2

–6 percentage units 1.2804 (0.1) 0.1365 0.76 1.1628 ns 0.1927 0.50 0.8243 ns 0.0644 0.60–3 percentage units 1.0500 ns 0.1349 0.61 0.9467 ns 0.1684 0.35 0.7990 (0.05) 0.0554 0.70Observed (31%) 0.9572 ns 0.0758 0.84 0.8602 ns 0.1091 0.63 0.7745 (0.05) 0.0563 0.67+3 percentage units 0.7304 (0.02) 0.0886 0.53 0.6468 (0.02) 0.1104 0.19 0.7455 (0.02) 0.0532 0.66+6 percentage units 0.7283 (0.02) 0.0777 0.67 0.6393 (0.02) 0.1050 0.31 0.7355 (0.02) 0.0498 0.70

Note: Mean parameter p and q values for whole root system were used. The simulated values are means of 75 runs of the FracRoot model. Allregressions for the root length were significant at P < 0.0001, for the number of links at P £ 0.0008, and for the root system mass at P £ 0.0002.SE is the standard error of the slope. The values in parentheses are P values for H0: slope = 1 in two-tailed Student’s t test. ns, nonsignificant.

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root system) produced an underestimation for the number oflinks within a root (Tables 4 and 5). The case that producesthe best estimate for the number of links should also resultin the best fit for root length if link length was accuratelyestimated. The current results suggest a small systematicoverestimation of root length. Further, quite a high variationof link length was observed in each branching generation ofP. deltoides (Fig. 3). Thus, stochastically calculating linklength using length distributions by link order and estimat-ing the length from them as we did here for parameters pand q is worth future research. However, that work will re-quire a larger number of link length measurements bybranching generation than we had available.

The root structure in FracRoot is currently static, providinga snapshot of the root architecture at the time of proximalroot measurements. Thus, the FracRoot model is not suitablefor simulating root development. Inclusion of the turnoverfactor may open possibilities for a dynamic root structuresimulation, i.e., an estimation of root system growth basedon diameter growth in proximal roots. However, the turnoverrate may vary with root conditions, such as diameter, rootdensity in the growing area, and availability of water and nu-trients (Jourdan and Rey 1997). Thus, repeated field meas-urements are needed for developing a dynamic rootstructure. Currently, these data are not available.

Ecological considerationsThe simulation results presented in Table 7 indicated that

several small proximal roots produce almost the same total

root system length as did fewer large proximals. This resultfollows from the mathematical structure of the FracRootmodel: because parameters p and q are ratios calculatedfrom squared root diameters, tapering of the root axes oc-curs at a faster rate in large diameters roots than in smalldiameter roots. This mathematical result is in agreementwith the lack of any trend in p and q values in relation tolink diameter (Table 1) and suggests that root diameterscales between links in a self-similar fashion (Mandelbrot1983; van Noordwijk et al. 1994) over the relatively limitedrange in our data.

The soil of the study site was quite uniform with very lowstoniness within the 1 m depth in which the whole root sys-tems were excavated. Uniformity was also reflected in thedistributions of the branching angles (Figs. 4 and 5), whichindicated very little soil restrictions to root growth — therewas a clear tendency for the daughter links to follow approx-imately the direction of the mother link. Much wider branch-ing angle distributions were observed in boreal forestsgrowing on more variable and much stonier soil (Kalliokoskiet al. 2008). This finding also suggests that the observed var-iability in link length may have been more affected by theinternal state of the trees than by soil conditions (Fig. 3).

The root system of P. deltoides appeared to be rela-tively deep compared with the horizontal extension(Fig. 8). This distribution may reflect the local weatherconditions — severe water deficits may develop in thefield site in late summer — and strong competition in thevery dense short-rotation plantation. The simulations

Table 7. Coarse-root system characteristics of five Populus deltoides trees simulated on the basisof proximal root data on each tree.

Tree code Npr* SD2pr (cm2){

Simulated length(m)

Simulated mass(kg)

Measured mass(kg)

RM 15 65.60 95.0 (91.0–99.0) 1.04 (1.01–1.07) 1.51WM 14 60.75 85.4 (81.4–89.5) 1.02 (0.99–1.05) 1.18WS 20 35.34 80.3 (77.5–83.1) 0.48 (0.47–0.49) 0.72YM 17 50.23 78.2 (75.3–81.1) 0.73 (0.71–0.75) 0.97YS 12 50.29 73.6 (70.7–76.5) 0.78 (0.76–0.80) 0.76

Note: The simulated values are means and 95% confidence interval of 75 runs of FracRoot model.*Number of proximal roots (measured).{Sum of squared diameters of proximal roots (measured).

Fig. 8. Visualization of the root system of a 4-year-old Populus deltoides growing in a dense short-rotation plantation; (a) top view at about198 angle, and (b) side view showing vertical distribution of the root system. Total coarse-root system length of the tree shown (YM) in thissimulation is 77.8 m and total number of links is 389. Horizontal and vertical scales are the same in both figures.

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agreed with the field observation that only single roots, ifany, had grown below 1 m depth. Although the horizontalextension of the root system of P. deltoides was limited(Fig. 8), root systems of the trees in the dense study standoverlapped strongly.

ConclusionsWhile some variability was observed in the scaling param-

eter p and allocation parameter q values within diameterclasses, the internal variability of root systems of P. del-toides had quite a small effect on the coarse-root length si-mulated using the FracRoot model compared with the effectof the root turnover factor. Including stochastic properties ofp and q and root turnover together in the model did not im-prove the simulation accuracy compared with the use of theturnover factor with invariant parameter values for the wholeroot system. The latter case is the recommended model for-mulation because it is accurate yet simple. Although includ-ing stochasticity of p and q did not improve modelbehaviour, stochasticity of link length is worth studying forfurther improvements of the model. Root length was muchmore sensitive to parameter values and turnover factor thanroot mass. Root length is, thus, the recommended metric formodel validation. The model validation results indicated thatFracRoot model gave poor estimates for individual roots butroot system architecture was relatively well described.

Coarse-root turnover has received relatively little attentionin root research. However, both our field observations andthe importance of the turnover factor to simulation accuracysuggest that coarse-root turnover should receive serious con-sideration, at least under conditions of strong competition orother severe external stress.

AcknowledgementsWe thank Kristopher Klaus and Daniel Gibbins for help

in root excavations and Prof. Stephen G. Pallardy for accessto his short-rotation experiment of P. deltoides. Fieldworkon P. deltoides was conducted while P.N. was working withthe Department of Forestry of the University of Missouri-Columbia with funding from the University of MissouriLife Sciences Mission Enhancement Program. His researchat the University of Helsinki was supported by grant No.210875 from the Academy of Finland. Contribution of M.L.was funded through the University of Missouri Center forAgroforestry under cooperative agreement AG-02100251with the USDA Agricultural Research Service (ARS) DaleBumpers Small Farms Research Center, Booneville, Arkan-sas. The results presented are the sole responsibility of thecoauthors and may not represent the policies or positions ofthe ARS.

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