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% Supplementary Information

# Supplementary methods
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We developed the equation of $\alpha_{c,f} = \alpha_{0,f} - \alpha_t \, t_f + \alpha_e \, t_c + \alpha_s \, \vert t_c-t_f \vert$ along with the basal area of each competitive species in the competition index to show the parameters are directly related to community weighted means of the different traits variables as:
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\begin{equation} \label{alphaBA}
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\sum_{c=1}^{N_i} {\alpha_{c,f} B_{i,c,p,s}} = \alpha_{0,f} \, B_{i,tot} - \alpha_t \, t_f \, B_{i,tot} + \alpha_e \, B_{i,t_c} + \alpha_s \, B_{i,\vert t_c - t_f \vert}
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\end{equation}
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Where:

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$B_{i,tot} = \sum_{c=1}^{N_i} {B_{i,c,p,s}}$,
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$B_{i,t_c} = \sum_{c=1}^{N_i} {t_c \times B_{i,c,p,s}}$,
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$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{N_i} {\vert t_c - t_f \vert \times B_{i,c,p,s}}$,
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and $N_i$ is the number of species in the local neighbourhood of the tree $i$.
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## Details on data sets used

Two main data type were used: national forest inventories data -- NFI, large permanent plots data -- LPP.
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### Panama

- Data set name: Panama
- Data type: LPP
- Plot size: 1 to 50 ha
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- Diameter at breast height threshold: 1 cm
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- Number of plots: 42
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: local
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- Contact of person in charge of data formatting: Plot data: R. Condit (conditr@gmail.com),  Traits data: J. Wright (wrightj@si.edu)
- Comments: The data used include both the 50 ha plot of BCI and the network of 1 ha plots from Condit et al. (2013). The two first census of BCI plot were excluded.
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- References: 
	- Condit, R. (1998). Tropical forest census plots. Springer, Berlin, Germany.
	- Condit, R., Engelbrecht, B.M.J., Pino, D., Perez, R., Turner, B.L., (2013).  Species distributions in response to individual soil nutrients and seasonal drought across a community of tropical trees. Proceedings of the National Academy of Sciences 110: 5064-5068.
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	- Wright, S.J., Kitajima, K., Kraft, N.J.B., Reich, P.B., Wright, I.J., Bunker, D.E., Condit, R., Dalling, J.W., Davies, S.J., DŐaz, S., Engelbrecht, B.M.J., Harms, K.E., Hubbell, S.P., Marks, C.O., Ruiz-Jaen, M.C., Salvador, C.M. & Zanne, A.E. (2010) Functional traits and the growth-mortality trade-off in tropical trees. Ecology 91: 3664-3674.
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### Japan

- Data set name: Japan
- Data type: LPP
- Plot size: 0.35 to 1.05 ha
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- Diameter at breast height threshold: 2.39 cm
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- Number of plots: 16
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: local
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- Contact of person in charge of data formatting: Plot data: M. I. Ishihara (moni1000f_networkcenter@fsc.hokudai.ac.jp),  Traits data: Y Onoda (yusuke.onoda@gmail.com)
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- Comments: 
- References: 
	- Yakushima Forest Environment Conservation Center, Ishihara, M.I., Suzuki, S.N., Nakamura, M., Enoki, T., Fujiwara, A., Hiura, T., Homma, K., Hoshino, D., Hoshizaki, K., Ida, H., Ishida, K., Itoh, A., Kaneko, T., Kubota, K., Kuraji, K., Kuramoto, S., Makita, A., Masaki, T., Namikawa, K., Niiyama, K., Noguchi, M., Nomiya, H., Ohkubo, T., Saito, S., Sakai, T., Sakimoto, M., Sakio, H., Shibano, H., Sugita, H., Suzuki, M., Takashima, A., Tanaka, N., Tashiro, N., Tokuchi, N., Yoshida, T., Yoshida, Y., (2011). Forest stand structure, composition, and dynamics in 34 sites over Japan. Ecological Research 26: 1007-1008. 


### Puerto Rico

- Data set name: Luquillo
- Data type: LPP
- Plot size: 16 ha
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- Diameter at breast height threshold: 1 cm
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- Number of plots: 1
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: local
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- Contact of person in charge of data formatting: Plot data: J. Thompson (jiom@ceh.ac.uk) and J. Zimmerman (esskz@ites.upr.edu),  Traits data: N. Swenson (swensonn@msu.edu )
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- Comments: 
- References: 
	- Thompson, J., N. Brokaw, J. K. Zimmerman, R. B. Waide, E. M. Everham III, D. J. Lodge, C. M. Taylor, D. GarciaMontiel, and M. Fluet. (2002). Land use history, environment, and tree composition in a tropical forest. Ecological Applications 12: 1344-1363.
	- Swenson, N.G., J.C. Stegen, S.J. Davies, D.L. Erickson, J. Forero-Montana, A.H. Hurlbert, W.J. Kress, J. Thompson, M. Uriarte, S.J. Wright and J.K. Zimmerman. (2012). Temporal turnover in the composition of tropical tree communities: functional determinism and phylogenetic stochasticity. Ecology 93: 490-499.


### Central African Republic

- Data set name: M'Baiki
- Data type: LPP
- Plot size: 4 ha
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- Diameter at breast height threshold: 10 cm
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- Number of plots: 10
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- Traits: Wood density and SLA
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- Source trait data: local
- Contact of person in charge of data formatting: G. Vieilledent (ghislain.vieilledent@cirad.fr)
- Comments: 
- References: 
	- Ouadraogo, D.-Y., Mortier, F., Gourlet-Fleury, S., Freycon, V., and Picard, N. (2013). Slow-growing species cope best with drought: evidence from long-term measurements in a tropical semi-deciduous moist forest of Central Africa. Journal of Ecology 101: 1459-1470.
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	- Gourlet-Fleury, S., V. Rossi, M. Rejou-Mechain, V. Freycon, A. Fayolle, L. Saint-Andr_, G. Cornu, J. Gerard, J. M. Sarrailh, and O. Flores. (2011). Environmental Filtering of Dense-Wooded Species Controls above-Ground Biomass Stored in African Moist Forests. Journal of Ecology 99: 981-90.
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### Taiwan

- Data set name: Fushan
- Data type: LPP
- Plot size: 25 ha
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- Diameter at breast height threshold: 1 cm
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- Number of plots: 1
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- Traits: Wood density and SLA
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- Source trait data: local
- Contact of person in charge of data formatting: I-F. Sun (ifsun@mail.ndhu.edu.tw)
- Comments: 
- References: 
	- Lasky, J.R., Sun, I., Su, S.-H., Chen, Z.-S., and Keitt, T.H. (2013). Trait-mediated effects of environmental filtering on tree community dynamics. Journal of Ecology 101: 722-733.


### French Guiana

- Data set name: Paracou
- Data type: LPP
- Plot size: 6.25 ha
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- Diameter at breast height threshold: 10 cm
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- Number of plots: 15
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- Traits: Wood density and SLA
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- Source trait data: local
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- Contact of person in charge of data formatting: Plot data: B. Herault (bruno.herault@cirad.fr),  Traits data: C. Baraloto (Chris.Baraloto@ecofog.gf)
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- Comments: 
- References: 
	- Herault, B., Bachelot, B., Poorter, L., Rossi, V., Bongers, F., Chave, J., Paine, C.E., Wagner, F., and Baraloto, C. (2011). Functional traits shape ontogenetic growth trajectories of rain forest tree species. Journal of Ecology 99: 1431-1440.
	- Herault, B., Ouallet, J., Blanc, L., Wagner, F., and Baraloto, C. (2010). Growth responses of neotropical trees to logging gaps. Journal of Applied Ecology 47: 821-831.
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	- Baraloto, C, P.C.E. Timothy, L. Poorter, J. Beauchene, D. Bonal, AM Domenach, B. H_rault, S. PatiĐo, JC Roggy, and Jerome Chave. (2010). Decoupled Leaf and Stem Economics in Rain Forest Trees. Ecology Letters 13: 1338-47.
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### France

- Data set name: France
- Data type: NFI
- Plot size: 0.017 to 0.07 ha
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- Diameter at breast height threshold: 7.5 cm
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- Number of plots: 41503
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
- Contact of person in charge of data formatting: G. Kunstler (georges.kunstler@gmail.com)
- Comments: The French NFI is based on temporary plot, but 5 years tree radial growth is estimated with short core. All trees with dbh > 7.5 cm, > 22.5 cm and > 37.5 cm were measured within a radius of 6 m, 9 m and 15 m, respectively. Plots are distributed over forest ecosystems on a 1-km 2 cell grid
- References: 
	- IFN. (2011). Les resultats issus des campagnes d'inventaire 2006, 2007, 2008, 2009, 2010 et 2011. Inventaire Forestier National, Nogent-sur-Vernisson, FR.
	- http://inventaire-forestier.ign.fr/spip/spip.php?rubrique153


### Spain

- Data set name: Spain
- Data type: NFI
- Plot size: 0.0078 to 0.19 ha
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- Diameter at breast height threshold: 7.5 cm
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- Number of plots: 49855
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
- Contact of person in charge of data formatting: M. Zavala (madezavala@gmail.com)
- Comments: Each SFI plot included four concentric circular sub-plots of 5, 10, 15 and 25-m radius. In these sub-plots, adult trees were sampled when diameter at breast height (d.b.h.) was 7.5-12.4 cm, 12.5-22.4 cm, 22.5-42.5 cm and >= 42.5 cm, respectively.
- References: 
	- Villaescusa, R. & Diaz, R. (1998) Segundo Inventario Forestal Nacional (1986-1996), Ministerio de Medio Ambiente, ICONA, Madrid.
	- Villanueva, J.A. (2004) Tercer Inventario Forestal Nacional (1997-2007). Comunidad de Madrid.  Ministerio de Medio Ambiente, Madrid.
	- http://www.magrama.gob.es/es/desarrollo-rural/temas/politica-forestal/inventario-cartografia/inventario-forestal-nacional/default.aspx


### Switzerland

- Data set name: Swiss
- Data type: NFI
- Plot size: 0.02 to 0.05 ha
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- Diameter at breast height threshold: 12 cm
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- Number of plots: 2665
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
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- Contact of person in charge of data formatting: M. Hanewinkel & N. E. Zimmermann (niklaus.zimmermann@wsl.ch)
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- Comments: All trees with dbh > 12 cm and > 36 cm were measured within a radius of 7.98 m and 12.62 m, respectively.
- References: 
	- http://www.lfi.ch/index-en.php
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	- Brandli, U.-B. (Red.) 2010: Schweizerisches Landesforstinventar. Ergebnisse der dritten Erhebung 2004-2006. Birmensdorf, Eidgenossische Forschungsanstalt fur Wald, Schnee und Landschaft WSL. Bern, Bundesamt fur Umwelt, BAFU. 312 S.
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### Sweden

- Data set name: Sweden
- Data type: NFI
- Plot size: 0.0019 to 0.0314 ha
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- Diameter at breast height threshold: 5 cm
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- Number of plots: 22904
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
- Contact of person in charge of data formatting: G. Stahl (Goran.Stahl@slu.se)
- Comments: All trees with dbh > 10 cm, were measured on circular plots of 10 m radius.
- References: 
	- Fridman, J., and Stahl, G. (2001). A three-step approach for modelling tree mortality in Swedish forests. Scandinavian Journal of Forest Research 16: 455-466.


### USA

- Data set name: US
- Data type: NFI
- Plot size: 0.0014 to 0.017 ha
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- Diameter at breast height threshold: 2.54 cm
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- Number of plots: 97434
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
- Contact of person in charge of data formatting: M. Vanderwel (Mark.Vanderwel@uregina.ca)
- Comments: FIA data are made up of cluster of 4 subplots of size 0.017 ha for tree dbh > 1.72 cm and nested in each subplot sapling plots of 0.0014 ha for trees dbh > 2.54 cm. The data of the four subplot were lumped together.
- References: 
	- http://www.fia.fs.fed.us/tools-data/


### Canada

- Data set name: Canada
- Data type: NFI
- Plot size: 0.02 to 0.18 ha
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- Diameter at breast height threshold: 2 cm
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- Number of plots: 15019
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: TRY
- Contact of person in charge of data formatting: J. Caspersen (john.caspersen@utoronto.ca)
- Comments: The protocol is variable between Provinces. A large proportion of data is from the Quebec province and the plot are 10 m in radius in this Province.
- References: 


### New Zealand

- Data set name: NZ
- Data type: NFI
- Plot size: 0.04 ha
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- Diameter at breast height threshold: 3 cm
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- Number of plots: 1415
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- Traits: Wood density, SLA, and Maximum height
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- Source trait data: local
- Contact of person in charge of data formatting: D. Laughlin (d.laughlin@waikato.ac.nz)
- Comments: Plots are 20 x 20 m.
- References: 
	- Wiser, S.K., Bellingham, P.J. & Burrows, L.E. (2001) Managing biodiversity information: development of New Zealand's National Vegetation Survey databank. New Zealand Journal of Ecology, 25: 1-17.
	- https://nvs.landcareresearch.co.nz/


### Australia

- Data set name: NSW
- Data type: NFI
- Plot size: 0.075 to 0.36 ha
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- Diameter at breast height threshold: 10 cm
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- Number of plots: 30
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- Traits: Wood density, and Maximum height
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- Source trait data: local
- Contact of person in charge of data formatting: R. M. Kooyman (robert@ecodingo.com.au)
- Comments: Permanents plots established by the NSW Department of State Forests or by RMK
- References: 
	- Kooyman, R.M. and Westoby, M. (2009) Costs of height gain in rainforest saplings: main stem scaling, functional traits and strategy variation across 75 species. Annals of Botany 104: 987-993.
	- Kooyman, R.M., Rossetto, M., Allen, C. and Cornwell, W. (2012) Australian tropical and sub-tropical rainforest: phylogeny, functional biogeography and environmental gradients. Biotropica 44: 668-679.

# Supplementary discussion

## Trait effects and potential mechanisms

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The most important driver of individual growth was individual tree size with a positive effect on basal area growth (see Extended data Table 3). This is unsurprising as tree size is known to be a key driver of tree growth[@Stephenson-2014; @Enquist-1999]. Then there was a consistent negative effect of the total basal area of neighbouring competitors across all biomes. The dominance of a competitive effect for the growth of adult trees (diameter at breast height >= 10cm diameter breast height) agrees well with the idea that facilitation processes are generally limited to the regeneration phase rather than to the adult stage [@Callaway-1997]. The variation of $\alpha_0$ between biomes is limited with large overlap of their confidences intervals.
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In terms of traits effects, wood density (WD) was strongly negatively associated with maximum growth, which is in agreement with the idea that shade-intolerant species with low wood density have faster growth in absence of competition (in full light conditions) than shade tolerant species[@Nock-2009; @Wright-2010]. One advantage of low wood density is clearly that it is cheaper to build light than dense wood, thus for the same biomass growth low wood density species will have higher basal area increments than species with high wood density[@Enquist-1999]. Other advantages of low wood density may include higher xylem conductivity[@Chave-2009], though for angiosperms this is a correlated trait rather than an direct consequence. A countervailing advantage for high wood density species was their better tolerance of competition (less growth reduction per unit of basal area of competitors), which is in line with the idea that these species are more shade tolerant[@Chave-2009; @Nock-2009; @Wright-2010]. This has generally been related to the higher survival associated with high wood density[@Kraft-2010] via resistance to mechanical damage, herbivores and pathogens[@Chave-2009; @Kraft-2010]. Yet this may also be related to lower maintenance respiration[@Larjavaara-2010]. For growth, lower respiration may lead to a direct advantage in deep shade, but this relationship might also arise through correlated selection for high survival and high growth in shade. Finally, high wood density was also weakly correlated with stronger competitive effects, especially in tropical forest where the confidence interval did not include zero. This might possibly have been mediated by larger crowns (both in depth and radius)[@Poorter-2006a; @Aiba-2009], casting a deeper shade.
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SLA was positively correlated with maximum basal area growth (growth without competition). This agrees well with previous studies that reported a positive correlation between SLA and gas exchange (the 'leaf economic spectrum'[@Wright-2004]). As in previous studies[@Poorter-2008; @Wright-2010], this direct effect of SLA was smaller than the effect size of wood density and had wider confidence intervals. Low SLA was also correlated with a stronger competitive effect. This may be related to a longer leaf life span characteristic of low SLA species because leaf longevity leads to a higher accumulation of leaf in the canopy and thus a higher light interception[@Niinemets-2010].
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Maximum height was weakly positively correlated with maximum growth rate (confidence intervals spanned zero except for temperate rain forest). Previous studies[@Poorter-2006a; @Poorter-2008; @Wright-2010] found mixed support for this relationship. Possible mechanisms are contradictory: maximum height may be associated with greater access to light and thus faster growth, but at the same time life history strategies might select for slower growth in long-lived plants[@Poorter-2008]. Maximum height was negatively correlated with tolerance to competition (confidence intervals spanned zero except for temperate rain forest and taiga), in line with the idea that sub-canopy trees are more shade-tolerant[@Poorter-2006a]. There was however a tendency for species with tall maximum height to have stronger competitive effects (though with wider confidence intervals intercepting zero). This might be explained by greater light interception from taller trees.
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Our results raise the question whether there is a coordination between trait values conferring strong competitive effect and trait values conferring high competitive tolerance. Competitive effect and tolerance are the two central elements of a species' competitive ability[@Goldberg-1991]. One may expect that because of intra-specific competition, species with strong competitive effects should have evolved a high tolerance to competition. However, in agreement with previous studies[@Goldberg-1990; @Goldberg-1991; @Wang-2010], we found little evidence for such coordination. There was only such a tendency for wood density and SLA. High wood density conferred better competitive tolerance and also stronger competitive effects, but with wide confidence intervals intercepting zero for the latter. High SLA conferred stronger competitive effects and higher tolerance of competition, but with wide confidence intervals intercepting zero for the latter. For maximum height, as explained above, there was a tendency for short maximum height to lead to high tolerance of competition but to low competitive effects. This is interesting because a trade-off between competitive tolerance and maximum height has been proposed as a fundamental mechanisms of coexistence of species in size-structured population in the stratification theory of species coexistence[@Kohyama-2009]. Finally the lack of support for coordination between tolerance and effects is important because it means that competitive interactions are not well described as a trait hierarchy relating a focal species to its competitors (measured as $t_c -t_f$ and thus assuming $\alpha_e = \alpha_t$ as in @Kunstler-2012; @Kraft-2014; @Lasky-2014). Traits of competitors alone or of focal plants alone may convey more information than the trait hierarchy. These processes of traits linked to either competitive effects or competitive tolerance, nevertheless, still lead to some trait values having an advantage in competitive interactions.
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Given that the effect sizes we report for effects of traits on competitive interactions are modest, the question arises whether the three traits available to us (wood density, SLA, and maximum height) were the best candidates. It is possible that traits more directly related to mechanisms of competition -- for instance for competition for light, the leaf area index of the competitors or the light compensation point at leaf or whole-plant level -- may be more powerful. It is also possible that traits measured at the individual level rather than as species averages might strengthen the predictive power of our analysis[@Kraft-2014].
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## Variations between biomes

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Overall, most results were rather consistent across biomes (Fig 2 main text), but some exceptions deserve comments.
For SLA, the sign of the tolerance of competition parameters were changing a lot among biomes (Fig. 2 main text). High SLA species tended to be more competition-tolerant (tolerance to competition parameter $\alpha_t$) in temperate forests (confidence interval only marginally intercepted zero) while low SLA species were more competition-tolerant in tropical forests. These different outcomes may trace to the prevalence of deciduous species in temperate forests (see Extended data Table 1), because the link between shade-tolerance and SLA is different for deciduous and evergreen species[@Lusk-2008]. In tropical forests shade-tolerant species often have long leaf lifespans, associated with low SLA. On the other hand in temperate deciduous forests the length of the growing season is fixed by temperature. Shade tolerant species cannot increase leaf longevity and instead reduce the cost of leaf production (high SLA) to offset the reduced income due to low light availability. The other noticeable difference between biomes was for taiga where the parameter relating wood density to competitive impact was positive, versus negative in the other biomes (Fig 2 main text). We do not have a mechanistic explanation for this discrepancy, but observe that taiga has relatively few species, many of which are conifers where the range of wood density is narrower than in angiosperms (see Extended data Table 1).
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# References