Báo cáo lâm nghiệp: "Modelling branchiness of Corsican pine (Pinus with mixed-effect models nigra Arnold ssp. laricio (Poiret) Francis"

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Original article Modelling branchiness of Corsican pine with mixed-effect models nigra Arnold ssp. laricio (Poiret) (Pinus Maire) b Colin Jean-Christophe Hervé Céline Meredieu Francis c Division a écosystèmes forestiers et paysage, Cemagref, domaine des Barres, 45290 Nogent-sur-Vernisson, France Unité b croissance, production et qualité du bois, Inra Nancy, 54280 Champenoux cedex, France. Équipe dynamique des systèmes forestiers, Engref/Inra, 14, rue Girardet, c 54042 Nancy cedex, France (Received 20 May 1997; accepted 15 December 1997) Abstract - In order to predict the influence of silvicultural practices on knottiness and canopy struc- ture, models were constructed to predict branch characteristics of Corsican Pine (Pinus nigra Arnold ssp. laricio (Poiret) Maire var. corsicana) from usual tree measurements. Thirty-six trees were sampled from a broad range of age, stand density and site index. Mixed models were fitted on branch length and on branch angle after a linearization procedure. A segmented second order polynomial model was chosen for the diameter profile. Nevertheless, this model seemed difficult to be transformed for fitting mixed models, but a global model was however satisfactory. For real- istic simulations with these models, we generated simulated correlated residuals of angle and diameter in order to maintain the negative link between angle and diameter of a branch. The mixed model method provides an improvement of the fit by accounting for inter-tree differences and/or intra-tree similarities and also an improvement of the simulation. The possibility to fit mixed- effect non-linear models allows a great number of models. (© Inra/Elsevier, Paris.) branchiness / Pinus nigra ssp. laricio / modelling / mixed model / simulation Résumé - Modélisation de la branchaison du pin laricio à l’aide de modèles à effets mixtes (Pinus nigra Arnold ssp. laricio (Poiret) Maire). Dans le but final de prévoir l’influence de la sylviculture sur la nodosité et le couvert, des modèles de prédiction des principales caractéristiques des branches à partir des données dendrométriques usuelles ont été construits. Trente-six pins lari- cio (Pinus nigra Arnold ssp. laricio (Poiret) Maire var. corsicana) ont été décrits. Les modèles * Correspondence and reprints Inra, Unité de Recherches forestières Méditerranéennes, Vivaldi, 84000 Avignon France. avenue E-mail: meredieu@avignon.inra.fr
mixtes ont été ajustés sur les données de longueur de branche et après linéarisation sur les don- nées d’angle de branche. Pour les diamètres, nous avons choisi un modèle polynomial de second ordre segmenté. Cependant, la méthode d’ajustement par la procédure de modèle mixte n’a pas pu être utilisée pour ce modèle complexe d’ajustement. Néanmoins un modèle global satisfaisant pour l’ensemble des arbres a pu être trouvé. Pour une simulation plus réaliste des angles et des dia- mètres des branches, nous avons généré des résidus corrélés afin de tenir compte de la liaison néga- tive qui existe entre angle et diamètre d’une branche. En conclusion, les modèles mixtes améliorent les ajustements en tenant compte des différences et/ou des similarités inter- et intra-arbre. La possibilité d’ajuster des modèles non linéaires à effets mixtes permet d’augmenter le choix des modèles. À partir de ces modèles les simulations sont améliorées. (© Inra/Elsevier, Paris.) nigra ssp. laricio / modélisation / modèle mixte / simulation branchaison / Pinus is the angle, the more important is 1. INTRODUCTION acute the volume of wood affected by the knot. The capacity to predict these branch char- Both silviculturists and wood scientists acteristics is therefore the key to the suc- need management tools that integrate both cessful linking of silvicultural regimes and growth and wood quality information in wood product quality [21]. Concerning order to predict the influence of site qual- length, extension and inclination of branch ity and silvicultural regime on individual this information is interesting for repre- tree growth, on stand yield, on stand struc- sentation of trees in the stand and of the ture and on wood quality. Our project con- canopy structure [5]. The branch length sists in setting up interconnected sub-mod- is important for the foliage display and els that range from growth models to saw thus influence tree growth [24]. The quality simulators. The quality is defined branch growth depends on the apical con- mainly in terms of wood density, mechan- trol of the main stem. It decreases from ical properties and knottiness. Directly the top of the tree to the base and from the linked with this latter property, the crown inside of the tree to the outside [11]. is considered to be the most sensitive com- ponent of a tree under direct influence of Numerous relationships between stem genetics and of surrounding growth con- and branches have been established on ditions [10, 31]. It is considered as a link other coniferous species, for instance: rela- between the internal structure of the stem, tionships between diameter at breast height the knottiness and the branch growth [4]. (DBH) and height, and the diameter of the thickest branch of Pinus sylvestris or of The branching of a tree must be speci- Picea abies [8], relationships between fied in terms of vertical profiles of branch DBH and the diameter of the thickest number per annual shoot, status: living or branch at 70 % of the height (Picea abies, dead, angle and diameter at insertion on [9]), DBH and height and mean branch the trunk, length, lateral extension and diameter at the base of the living crown inclination of each whorl branch along the [ 19]. Mitchell [23] observed that for Picea stem. The first four variables are consid- glauca the length of whorl branch at the ered as inputs into wood quality simula- largest extension of the crown is related tors. These simulators have a prediction with the length between the tree top and component of knot shape (for instance the branch insertion. Considering these Leban and Duchanois [14]). This shape is previous results we planned to predict the predictable if branch angle and diameter as vertical profiles of branch characteristics well as bole radius are available. It has based on the stem attributes (such as DBH, been shown for instance that the more
In order to fit these profiles, height, etc.). methods have been proposed [4] but they still need to be improved to clearly dis- tinguish different levels of variability and between-tree variability. Therefore, the aim of this paper is to propose the use of mixed models which improves both modelling and simulation of the vertical profiles of angle, diameter and length of branch for Corsican pine. 2. MATERIALS AND METHODS 2.1. Structural development of Corsican pine This study concerns Corsican pine (Pinus nigra Arnold ssp. laricio (Poiret) Maire var. It is fast growing conifer with corsicana). a and a straight stem. The good wood quality the main stem, branches directly inserted on stem and the branches are monopodial with a the primary branches. strong apical control. Branch and stem elon- gation are annual, and each annual shoot con- stitutes one growth unit (GU). Lateral branch- ing usually occursI year after bud formation 2.3. Variables (proleptic branching). Branches are located only at the top of each annual growth unit, clus- tered in whorls. There are no inter-whorl Two kinds of variables were used: branch branches. and whole tree descriptors (table descriptors II). The latter were the standard tree measure- ments, crown heights and combination of vari- ables: diameter at breast height (DBH) (in cm, 2.2. Sampling measured to the nearest 5 mm); total height (TH) of the stem (in m, to the nearest 5 cm); Twelve plantations of Corsican pine located height to the lowest living branch (HLLB) (in in the region ’Centre’ (France) were selected in m, to the nearest 5 cm); height to the lowest order to investigate a broad range of age, indi- living whorl (HLLW) (in m, to the nearest vidual growing space and site index (table I). 5 cm) where at least three quarters of branches We tried to select only trees from the variety living; crown length 1 (CL): were TH - HLLB (in m); crown length 2 corsicana. Artificially pruned plantations and CL = unhealthy plantations were avoided. In each (HLLWA): HLLWA TH - HLLW (in m); = plantation, a 0.3-ha plot was installed. Girth length between the lowest living whorl and the at breast height of all trees was measured in lowest living branch (DIFF) (in m); crown ratio order to select three trees of different social 1 (CR): CR (TH - HLLW)/TH; crown = status: one dominated (from the lower girth ratio 3 (CR3): CR3 = (TH - HLLB)/TH; crown class), one medium (with a girth near the mean ratio 2 (CR2): CR2 = (CR + CR3)/2; based on girth) and one dominant (from the higher girth the bud-scale scars and whorl, each annual elongation or growth unit (GU) was identified class). Trees with very serious crown defects by its rank recorded from the tree top: GU =I and with forked stems were excluded. Thirty- six trees were felled and measured in 1995. is the youngest growth unit elongated in year y, GU 2 is the growth unit elongated in y - 1, Data describing crown size and development = were collected, based on the analysis of all the etc.; length of each annual growth unit
2.4.1. First step: establishment of (GUlength) measured between two successive whorls (in cm, to the nearest 5 cm); distance a global fixed-effect model from the apex (DFA): absolute distance from each whorl (top of the growth unit) to the top of First, we modelled the variation of each the stem (in m, to the nearest 5 cm); age (A) of variable along each stem with individual equa- the tree based on the number of whorls and tions (one per tree) [22] as: checked by counting the annual rings on the stump section after felling (in years). Y represents for example the branch where ij For each branch, we measured: branch angle, X the independent variable (i.e. the pre- ij diameter over bark (BD) (in mm, to the nearest dictors),i denotes the ith tree, j the jth annual mm, at a distance from the bole that was growth unit, &thetas; model parameters specific to i the approximately equal to one branch diameter, on the ith tree and &i,j the within-tree residual vari- epsiv; the horizontal axis; dead and living branches ation. are measured); branch insertion angle (BA) (in Then, linear regressions were carried out grad, to the nearest 5 grad) between the axis in order to analyse the variability of the param- of the branch and the axis of the main stem eters &i; in relation to the whole tree descrip- thetas (dead and living branches are measured); tors: branch length (BL) (in cm, to the nearest I cm): distance between the top of the branch and its branch insertion (living branches are measured). &iksateht ; denotes the kth parameter in the vec- where &thetas; denotes the global k . ψ i tor parameters com- 2.4. Statistical analysis sampled trees and &ik is a random all mon to eta; error representing unexplained between-tree variations. It may happen that some parame- The variables to be predicted are BL, BA ters &ik; do not have any clear relationship with thetas and BD, as functions of DFA, GU and/or the whole tree descriptors; in this case, equation GUlength and also whole tree descriptors. The (2) above simply becomes: &ik; = μ + &ik where thetas k eta; analyses were carried out following three main μ is the mean of &ik; over the trees. thetas k steps.
Finally, we transformed the individual mod- where &thetas; is the mean of &isateht ; over the trees and &iatled ; els into a global model by progressively replac- is the ith tree individual deviation from &thetas;, with ing the &isateht ; parameters in (1) by their predictions mean zero and variance-covariance matrix B. (2). We obtained fixed-effect models of the Returning to Y this yields: , ij following form: where (1) is explained variation (fixed effects), In such models, the unexplained between- (2) is the between-tree random deviation (tree tree variations η in (2) are neglected. ik effects), and (3) is the within-tree residual vari- ation. 2.4.2. Second step: linearization of the In general, the tree vector parameters &isateht ; global model, if necessary may also be a linear function of tree variables, and some parameters in &isateht ; may have no ran- In order to change the global models into dom variation. The general mixed-effect model mixed models, we had to transform the non- is then: linear models into linear ones. Several methods have been proposed to fit non-linear mixed models (Lindstrom and Bates [16]; Wolfinger where &thetas; is a fixed-effect parameter vector, X ij [32]; and others: see Gregoire and Schaben- is the fixed-effect variables, b the vector of iis berger [7]). One of the simplest due to Beal random effects, Z is the vector of vari- (tree) ij and Sheiner [2] and used by Gregoire and Sch- ables associated with the random effects, ϵijis abenberger [7] is to approximate the marginal the residual within-tree random variation. Fur- distribution of the response vector by expand- thermore b and &ij; are supposed normally dis- i epsiv ing the global model in a first-order Taylor tributed, independently of each others: series; it was adopted in the present study. 2.4.3. Third step: fitting of mixed linear model In this step, we improved the linear fixed- where B is the variance-covariance matrix of effect model by taking into account the vari- random effects and &2 the within-tree residual sigma; ability between trees due to the mixed model. variance. The mixed linear model (with both fixed The unknown parameters (fixed-effect and random effects) is a generalization of the parameter &thetas;, variances and covariances of ran- standard linear model; the generalization being dom effects and residual variance &2 are esti- ) sigma; that it is possible to analyse data with several mated using restricted maximum likelihood sources of variation and especially within- and between-tree variations. For a detailed descrip- [13]. tion of the theory, refer to Laird and Ware [13] One of the criteria to measure the goodness and Diggle et al. [6]. of fit is the Akaike information criterion (AIC) Suppose for example that a variable Y (for ij which permits us to compare models with the instance branch length) is linearly linked to same fixed effects but different variance struc- others variables X within each tree: &jadnr; + 2 p ij ture. AIC is computed as: AIC = where &jadnr; is twice the negative log-likelihood evaluated at the maximum likelihood estimates where the within-tree unexplained vari- a&ij nd s epsiv;’ i (or restricted maximum likelihood estimates) ation &i the vector of parameters, spe- thetas; is and p is the number of estimated parameters. cific of the ith tree. Suppose that &isateht ; varies from A procedure was designed especially for independently at random, so that, tree to tree fitting mixed linear model in SAS/STAT. This when looking at different trees, one can write: function called PROC MIXED has been avail- able in release 6.07 [17, 28].
2.5. Generation of values from normal distribution with mean 0 and given variance-covariance matrix After improving each model and its fitting with our data, the last stage was to include them in a growth simulator. With the charac- teristics of simulated trees given by the growth simulator, our equations permit us to simulate their branchiness. For each equation con- structed with mixed model procedure, or in order to take into account the correlation between residuals of different models, we had to generate values of parameters (for example, γ1, γ2) from normal distribution with mean 0 and given variance-covariance matrix V: We first generated a random vector g from a normal distribution with mean 0 and identity variance-covariance matrix, using the SAS function Normal [28]. g is then transformed to a N (0, V) by multiplying it by a lower triangu- lar matrix L such that LL = V (Cholevski t decomposition of V). the apex the independent vari- (DFA) as 3. RESULTS able: For diameter and angle, dead and living branches were taken into account in the fit of individual profiles. In fact for each where aand a are the individual param- 2i 1i tree, the scatter of diameter and angle data eters for each tree. The R-square ranged is continuous at the level where the two between 0.93 and 0.99 and the residual kinds of branches overlap. Only living standard deviation ranged between 20 and branches were taken into account for 55 cm depending on trees. Then, we anal- branch length. For all the following anal- ysed the variability of the parameters a 1i yses, we removed branches with a partic- and a in relation to the whole tree 2i acute angle (ramicorn branches) ularly descriptors. DBH had a linear relationship (i.e. a total of seven branches in sampled with a (R 0.30). Other tree descrip- 1i 2 = trees) and branches with a diameter of less tors, such as crown variables, did not show than half of the maximum diameter per any relationship with the individuals whorl (figure 1).In term of knottiness, the parameters. This led to the following influence of these thin branches can be global model with DFA, DFA and 2 neglected. DFA.DBH as independent variables (fitted to1782 branches): 3.1. Branch length 3.1.1. Construction of a linear mixed model First, for each tree, we fitted individ- The R-square was equal to 0.835 and ual linear models with the distance from the residual standard deviation was equal
This fixed-effect model has been files of trees. For each tree, the values of 43 to cm. improved by taking into account the vari- α &1i and &2i were generated from a , 0i alpha; alpha; trivariate normal distribution with mean ability between trees due to the mixed lin- model. This was carried out here by 0 and variance-covariance matrix given ear adding a random tree component to some by the fit of model (4). Then for each branch in a tree, a random residual was or all parameters a a a a Among ,,,. 0123 several attempts of different random effect simulated from a normal distribution with mean zero and variance equal to the resid- combinations, we chose the model with the highest AIC, which included random ual variance of the model (4). Figure 2b effects for the intercept, DFA, DFA : 2 shows an example of a simulated branch length profile from growth characteristics (DFA, DBH, number of branches per whorl). Parameter estimates of fixed effects 3.2. Branch angle and diameter (A A A A variances/covariances 0, 1 2 3 , , ), of random effects (α α α are pro- ,,) 0i 1i 2i 3.2.1. The branch angle model: vided in table III. The standard deviation a non-linear mixed model was only 34.3 cm. Figure 2a shows how the mixed linear model fits the data for tree. To predict the vertical trend of angle one for both dead and living branches, we used 3.1.2. Simulation of branch length the number of annual growth units counted profile downwards from the top of the stem (vari- able GU) as independent variable. Accord- The estimates of the covariance param- ing to Meredieu and Colin [22], we fitted allowed us to simulate different pro- the following non-linear individual model: eters
where b1i b are parameters con- and 2i strained to be positive, and c is a constant (c=51). We did not obtain good relationships between individual parameters and global tree characteristics. Nevertheless, it was where the derivative expressions mean the important to characterize both inter- derivative with respect to the indi- partial individual and intra-individual variations cated parameter, evaluated at the guessed adequately in order to permit valid simu- values. We have: lations. For this reason, we adopted the following mixed-effect model: To fit this non-linear mixed model we approximated the marginal distribution of rearranging, this yields: and the response vector by expanding our model (6) in a first-order Taylor series. A first-order Taylor expansion of our model, around guessed values b and b10 20 (obtained by fitting the fixed-effect ver- Model (8) is a linear mixed model sion of model (6)), gave the following where the non-linear response has been approximate linear model (7): replaced by an approximated linear model.
joint value ξ which is the location of the The approximation is better if the guessed estimated thickest branch: values band b are chosen near the true 20 10 unknown values b and b We used esti- 1 . 2 mates obtained by fitting the fixed-effect version of model (6). Then we estimated the parameters of the new linear mixed- effect model. New estimates were obtained. If necessary we repeated this step with the updated estimates as guessed where d0, d1, d2, d3 and d4 are values until the change in the likelihood constrained parameters and positive insignificant. was ξ=-d1/(2 d2). The fixed effect version of (8) was fit- This model was fitted independently ted to benchmark the anticipated improve- for each tree [22]. Then we tried to con- ment offered by the inclusion of the ran- struct a single global model. First, d4 was dom parameters in the model. When (8) fixed at 1. This parameter was therefore was fitted with only fixed effects, transformed into a constant in further ana- AIC =-17003 and when it was fitted with lyses. Then, we analysed the variability random and fixed effects, AIC =-16535. of the parameters ξ, d1 and d3 in relation For this latter model &2sigma; 118.2 and all = to the whole tree descriptors. Among var- the parameters were clearly significant ious combinations of whole-tree descrip- (table IV). The branch angle profile pre- tors, the best fits were given by: dicted by this model is shown by a solid line in figure 3 and the deficiencies of the fixed-effect version of the model become apparent. 3.2.2. Construction of the branch diameter model: a fixed-effect model Consequently, we deduced from ξand d1 one relation for d2. We formed a global According to Colin and Houllier [3] we model (12) which was estimated for all chose for the diameter profile a segmented second order polynomial model with a trees.
had to take into account the link we between diameter and angle within a tree: globally, the thickest branches had an acute angle whereas the thinner branches had an open angle. Although a part of this link is taken into account in the models above through the common or linked vari- ables of each model, it remains a rela- tionship between the unexplained within- tree variations of diameter and angle (figure 5). The parameter values and their stan- dard errors were estimated as follows in Formally, for a branch j in a tree i, we table V (statistics of fit for 4 303 obser- have to simulate the branch angle BA ij vations on 36 trees: root mean square error and the branch diameter BD according ij (RMSE) 6.5 mm). Figure4 shows how = to: the model fits the data for two trees. If the linearization of this complex model was also possible using the same method as for branch angle, the mixed model introduced numerous fixed and ran- where f is the angle model (8), f the A D dom effects which could not be dealt with diameter model, B and d the fixed-effect D A classical statistical package (SAS statis- parameters for angle and diameter, respec- tical package). tively, &i; is the vector of random tree beta effects for angle (to be generated from a 3.2.3. Joint simulation of branch bivariate normal distribution with vari- length and branch angle profiles ance-covariance matrix as given in table IV, and ϵ and ϵ are the correlated Aij Dij To simulate diameter and angle pro- within-tree random variations of angle and files along a tree stem in a realistic fashion, diameter, jointly distributed as a bivari-
normal distribution with 0 and ate mean variance-covariance matrix: The fit of the angle and diameter mod- els provides estimates for var(ϵ and ) Aij var(ϵ (tables IV and V), but does not ) Dij for the correlation p between ϵ and ϵ Aij . Dij This would have required the joint fit of the diameter and angle models, which is possible in principle, but lead to a too com- plicated computation problem. Instead, we have estimated p by the observed cor- relation between the residuals of the indi-
vidual models of angle and diameter (fig- When we analysed individual models ure 5: for example for this tree, p= 0.66). of branch characteristics, we noted a data Figure 6a shows an example of simulated structure. The mixed models allow analy- values of branch diameter and angle along sis of data with several sources of varia- a tree stem; figure 6b illustrates the within- tion. Like the traditional linear model we can analyse the mean variability of sample tree correlation obtained between simu- lated diameters and angles. trees, but we can also share the residual variability in a between-tree variation and an unexplained individual variation. This method leads us to decrease the residual 4. DISCUSSION variability of the traditional linear model. This method can have a great interest in 4.1. Statistical methods forestry research where the between-tree variation for a variable is sometimes larger This paper gives information on how than the variation between populations. mixed models on branch length and to use Indeed in our model of branch length branch angle and how to generate vari- had a better estimation of the fixed we ability around the mean profiles of diam- effect due to the association of the tree eter and angle by taking the correlation of characteristics and an estimation of the the residuals into account. variability inside the sampled trees. We did not use the joint generalized 4.1.1. The proposed mixed models least-squares method (as explained by Meredieu and Colin [22]) which permitted the estimation of a system of independent Compared to other studies, the pro- posed models present an improvement equations of branch diameter and branch concerning the statistical description of angle with correlated random errors. But, the variability. we improved the model of branch angle
with a mixed model as well as the branch tions with which the outputs of a distance- diameter model. For this latter we found a independent growth model were able to global model. Furthermore, when we sim- give us detailed information about branch- iness. The role of the branch equations is ulated a tree we took into account the cor- to provide information first about branch- relation between the residuals of the two iness for technical pruning management models. and second about knottiness at different The possibility to fit mixed-effect non- simulation steps (thinning or clear cutting) linear models allows a great number of for logging operations and for grading models. Nevertheless, for complex models standing or felled trees. Furthermore our such as the segmented second order poly- goal is to provide a management tool nomial model, the linearization was pos- which permits to give information on yield sible, but the capacity of the statistical and wood quality with the minimum infor- package did not allow the fitting of this mation usually measured by forest man- type of model with too numerous vari- agers. Lastly, our model could be used to ables. predict the external shape of the crown with the branch length model. 4.1.2. The simulation of tree branchiness 4.2.2. Choice of tree descriptors Mixed models also have a great power The choice of tree descriptors was lim- for simulation: the fixed effect model gives ited by the tree attributes usually provided the mean tendency, the random effect by growth simulators such as tree age (A), model with the variance-covariance height (TH), past height growth curve, tree matrix gives several possible tendencies past height annual increments curve, DBH, within the population. We just have to past radial growth curve, height to the low- generate a vector ofpositions of mean 0 living whorl (HLLW), height to the est and given variance-covariance matrix of lowest living whorl branch (HLLB), and random effect. We made one drawing of a their combinations such as different crown value per tree and then one drawing of a ratios, etc. For a given growth unit we value for each branch in the have: its identification number GU and variance-covariance matrix of within- the distance to the apex of its upper limit branch error. (DFA). We did not use a system of equations We did not use stand variables such as for fitting both angle and diameter but we local density, basal area, stand density or took into account the residual correlation social status. We chose allometric rela- in the simulation. This method permits us tions between bole dimensions and crown to improve the characteristics of simulated dimensions. Silvicultural effects are taken trees used in a wood quality simulator. into account through the growth models The branch characteristics are now com- which give the characteristics of trees and patible thus the biggest branch of a whorl through the values of variables reflecting has one of the smallest angles. the past growth conditions. The mean vertical profile of length can 4.2. Methodological aspects be obtained, based on information on DFA static model 4.2.1. Choice of and DBH. This result is similar to the con- clusions of Mitchell [23] and Kozlowski The choice of static allometric models [11]concerning the relation between is due to several reasons: we need equa- branch length and its insertion level from
the apex. At the same distance from the real into a provenance variability and a apex, the increase of tree size (DBH) between-tree variability. induce the increase in branch length. Like 4.2.3. Sampling design and statistical Kramer [ 12] we showed that the branch models lifespan becomes longer as the tree ages because in the equation the increase in Our sampling design had several lev- DFA implies an increase in tree age. els of variability: between-stand (site, age, silvicultural and genetics effects), within- The mean vertical profile of diameter, stand (social status effects), within-tree model introduced by Colin and Houllier (i.e. between-whorl including age, climat [4], can be obtained based on information and varying-over-time tree to tree com- on DFA, DBH, CR2 and HLLWA. This petition effects) and within-whorl vari- model is consistent with other results: ability (within-branch competition and Schöpf [29], Maguire et al. [20] and Uus- position effects [15]). Our results have vaara [30] observed that the branch size taken into account the within-tree and a is linked to crown shape (CR2, HLLWA) between-tree variability. With the infor- and to stem size (DBH). Nevertheless it mation of residual variability it is possi- is known that branch diameter is under ble to include the within-whorl variabil- environmental control (for instance spac- ity. Our sampling describes a great ing) [1]. CR2 and HLLWA could give variability needed by the final objective indications on silvicultural management. of the study: the prediction of branchiness characteristics of simulated trees from a It was impossible to obtain such rela- growth model. tions between usual tree measurements To check if social status, age or fertility and branch angle profile. Nevertheless, had an effect, we used the random-effect there is an effect of tree age on the inser- solution vector of the branch length model tion angle of the branches. GU gives infor- and the branch angle model. With box- mation on the position of the branch in plots, we were able to verify the non- the crown: first, in the upper crown, angle effects of these sampling variables. So, is linked with the apical control and these effects were taken into account branches are oriented towards the light. through the tree characteristics used in Second, due to the mechanical pressure these mixed models. Nevertheless, we of the stem wood at the insertion point could not check the between-stand vari- and due to the branch weight the angle ability. Only three trees per stand were becomes larger. We did not observe any studied, and these trees had different social effect of silvicultural management through status. There are possibilities for improv- tree variables but there is a random effect ing the description of this variability due to between trees. This global result has how- silvicultural management by increasing ever to be considered according to the the sampling on some stands. genetic origin of the trees. It is known that branch angle is under stronger genetic con- We chose trees without crown defects. trol [30]. Comparative studies between The occurrence of ramicom branches were Corsican pine varieties (corsicana and cal- not be studied here. We know that it abrica) showed different angle patterns on the genetic origin, depends partially [25]. Comparative studies on different the site conditions and tree age [26, 27]. It provenances are needed to remove the can be interesting to study this phe- genetic effects from the random parame- nomenon more intensively in order to take ters of our model. It can be interesting to this aspect into account by stochastic sim- distinguish the variability between trees ulation at the stand scale.
The branches analysed for their angle the radial external limit of the bole. To and diameter are only branches with a rel- obtain the entire knot shape inside the bole, atively large diameter (i.e. branches we use a stem taper equation to obtain the exhibiting a diameter higher than half of bole radius at each level. The information the maximum diameter per whorl). The on branch angle, bole radius and distance from the apex may give indication on the removed branches have a low impact on wood quality. These thin branches have a curvature of the pith of the knot and with the branch diameter on the ovality of the short lifespan, are located at the base of the whorl and self-pruned after a short knot inside the bole. Then, it will be pos- time. This kind of branch represented from sible to describe the envelope of the knot. 2 to 20 % of all branches. Further studies Furthermore, we modelled the proportion on the removed branches should have been of living branch per whorl [22], so we are carried out in order to explain further bio- able to give information on the delineation logical aspects. In the branch length between tight and loose black knot zones model, we only studied living branches [18]. which give the crown shape. This work will finally lead to statistical relationships between branching traits, knot shape and also knot volume. The set 4.3. Evaluation of this type of model of branch models and the set of knot mod- els will be then combined. We will then be able to link tree attributes to knottiness Although the crown morphology results features, and also growth conditions to from dynamic functional processes which knottiness and crown structure. include genetics, site conditions, stage of development of the tree, local density of the stand, social status of the tree, our ACKNOWLEDGEMENTS models do not need information about the stand but only current tree characteristics. This research was initiated by Inra, Cema- So, for evaluation, we do not need to know IDF and was supported in parts by gref and the site quality nor the past silvicultural grants from AGRIDEM (1995). The authors management to choose sample trees. Con- are grateful to R. Chevalier (Cemagref), D. versely to growth models which need trees Riotord (Inra Avignon), W. Mirlyaz (IDF), J. belonging to stands with past information, Dulac and S. Mathieu for their technical help our branch models are easy to evaluate and D. Auclair, J. Chadoeuf and P. Dreyfus. with new data. The authors also thank the private owners and the Office National des Forêts (ONF) for autho- Mixed model interest for evaluation is rization to fell and measure their trees. allow us to check both if the mean ten- to dency is the same for the new sample (same fixed effects) and if the range of REFERENCES variation of new sample trees is well explained with the variance-covariance Long J.N., Influence of stand Ballard L.A., [1] log quality of Lodgepole pine, density matrix of random effects. on Can. J. For. Res. 18 (1988) 911-915. Beal S.L., Shciner L.B., The NONMEM sys- [21] tem, Am. Stat. 34 (1980) 118-119. 4.4. Vertical profile of knot size Colin F., Houllier F., Branchiness of Norway [3] spruce in north-eastern France: modelling and shape vertical trends in maximum nodal branch size, Ann. Sci. For. 48 (1991) 679-693. Branch diameter and angle at a given Colin F., Houllier F., Branchiness of Norway [4] height give access to the size of the knot at spruce in north-eastern France: predicting the
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