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% Supplementary Information
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# Supplementary methods
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We developed the equation of $\alpha_{c,f} = \alpha_{0,f,intra} \, CON + \alpha_{0,f,inter} \ (1-CON) - \alpha_t \, t_f + \alpha_e \, t_c + \alpha_s \, \vert t_c-t_f \vert$ 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,intra} \, B_{i,f} + \alpha_{0,f,inter} \, B_{i,het}- \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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$B_{i,het} = \sum_{c \neq f} {B_{i,c}}$,
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$B_{i,t_c} = \sum_{c=1}^{N_i} {t_c \times B_{i,c}}$,
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$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{N_i} {\vert t_c - t_f \vert \times B_{i,c}}$,

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and $N_i$ is the number of species in the local neighbourhood of the tree $i$. Note that the index $p$ and $s$ respectively for plot and data set were not include for sake of simplicity.
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## Derivation of $\rho$ for a Lotka-Volterra model based on Godoy \& Levine[@Godoy-2014]

Chesson[@Chesson-2012] proposed to estimate the stabilising niche difference, based on the per capita growth rate of rare invader in the population of a resident species. Godoy \& Levine[@Godoy-2014] used this method on an annual plant population model. This approach can be explained using teh Lotka-Volterra model, defined as:

\begin{equation}
\frac{dN_i}{dt} =  N_i \times r_i \times (1 - \alpha'_{ii} N_i -
\alpha'_{ij} N_j)
\end{equation}

The criteria for invasion of species $i$ in resident community of species $j$ is (at equilibrium population of single resident $j$ is $\overline{N_j} = \frac{1}{\alpha'_{jj}}$):

\begin{equation}
(1 - \frac{\alpha_{ij}}{\alpha'_{jj}})
\end{equation}

Thus if $\frac{\alpha_{ij}}{\alpha'_{jj}} <1$ invasion $i$ in $j$ is possible (same approach for $j$ in $i$).

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Stable coexistence for species $i$ in regard to speciesand $j$ thus requires
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$\frac{\alpha'_{ij}}{\alpha'_{jj}}$ and $\frac{\alpha'_{ji}}{\alpha'_{i}}$ to be smaller than 1. Chesson[@Chesson-2012] then defioned the average
stabilising niche
overlap between species $i$ and $j$ as:

\begin{equation}
\rho = \sqrt{\frac{\alpha'_{ij} \alpha'_{ji}}{\alpha'_{jj} \alpha'_{ii}}}
\end{equation}
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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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```{r kable, echo = FALSE, results="asis"}
library(plyr)
dat <- read.csv('../../data/metadata/sites/sites_description.csv', check.names=FALSE, stringsAsFactors=FALSE)
# reorder so references column is last
i <- match("References", names(dat))
dat <- dat[,c(seq_len(ncol(dat))[-c(i)], i)]
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dat <- dat[,c(2,1,3:ncol(dat))]
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refs <- read.csv('../../data/metadata/sites/references.csv', check.names=FALSE,
                  stringsAsFactors=FALSE)
refs$citation <- iconv(refs$citation, "ISO_8859-2", "UTF-8")
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replace_refs <- function(x){
	ids <- as.numeric(unlist(strsplit(x,",")))
	if(length(ids>0))
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		ret <-	paste0("\n\t- ", refs$citation[match(ids, refs$id)],
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                               collapse="")
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	else
		ret <- ""
}

dat$References <- sapply(dat$References, replace_refs)

paste_name_data <- function(df){
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	sprintf("### %s\n\n%s\n\n", df[["Country"]],
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		paste0(
			llply(names(df)[-c(1)], function(x) sprintf("- %s: %s", x, df[[x]])), collapse="\n")
		)
}
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list.t <- dlply(dat, 1, paste_name_data)
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writeLines(unlist(list.t[dat[["Country"]]]))
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```

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\newpage

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## Age distribution for Europe and North America.
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![Age distribution of forest area in 20-year age class for France, Switzerland and Sweden, estimated by Vilen et al.[@Vilen-2012]. The last class plotted at 150 years is for age > 140 years (except for Sweden where the last class 110 is age > 100 years).](../../figs/age_europe.pdf)

![Age distribution of forest area in 20-year age class for North America (USA and Canada), estimated by Pan et al.[@Pan-2011]. The last class plotted at 150 years is for age > 140 years.](../../figs/age_na.pdf)

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\newpage
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# Supplementary results

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![**Global trait effects and trait-independent effects on maximum growth and competition and their variation among biomes for a model with no differences in $\alpha_0$ for intra- and interspecific competition.** Standardized regression coefficients for growth models, fitted separately for each trait (points: mean estimates and lines: 95\% confidence intervals). Black points and lines represent global estimates and coloured points and lines represent the biome level estimates. The parameter estimates represent: effect of focal tree's trait value on maximum growth ($m_1$), the competitive effect independent of traits ($\alpha_{0}$), the effect of competitor trait values on their competitive effect ($\alpha_e$) (positive values indicate that higher trait values lead to a stronger reduction in growth of the focal tree), the effect of the focal tree's trait value on its tolerance of competition($\alpha_t$) (positive values indicate that greater trait values result in greater tolerance of competition), and the effect on competition of trait similarity between the focal tree and its competitors ($\alpha_s$) (negative values indicate that higher trait similarity leads to a stronger reduction of the growth of the focal tree). Tropical rainforest and tropical seasonal forest were merged together as tropical forest, tundra was merged with taiga, and desert was not included as too few plots were available (see Fig 1a. for biomes definitions).](../../figs/figres12_TP.pdf)


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![**Global trait effects and trait-independent effects on maximum growth and competition and their variation among biomes for a model with random structure in the parameter including the data set and the Koppen-Geiger ecoregion.** see the figure 2 in the main text for parameters description and see Fig 1a. in the main text for biomes definitions.](../../figs/figres12_ecocode_TP_intra.pdf)
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\newpage
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# Supplementary discussion

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## 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 conspecific and heterospecific 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 $\alpha_{0 \, intra} \, \& \, \alpha_{0 \, inter}$ 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. 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 only in three biomes and with a weak effect. Previous studies have reported a strong positive correlation between SLA and gas exchange (the 'leaf economic spectrum'[@Wright-2004]) and young seedling relative growth rate (REF TO ADD). Studies[@Poorter-2008; @Wright-2010] on adult tree have however generally reported weak and marginal correlation between SLA and maximum growth. Our results support this pattern. 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. These small effect of maximum height are probably explained by the fact that our analysis focus on short-term effect on tree growth. Size-structure population models[@Adams-2007] have in contrast shown that maximum height is a key drivers of the long-term competitive success in term of population growth rate.
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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. We found clear evidence for such coordination for wood density, but not for the other traits. High wood density conferred better competitive tolerance and also stronger competitive effects. High SLA conferred stronger competitive effects but not effect on tolerance of competition. For maximum height, as explained above, there was a tendency for short maximum height to lead to high tolerance of competition (see also the Figure 4 in Supplementary Results), but no link with competitive effect. The mixed results on the coordination between tolerance and effects are important because they mean 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. It is also important to not that an analysis that do not account for the trait independent differences between intraspecific *vs.* interspecific led to an overestimation of the trait similarity effect (Figure 3 in Supplementary results).
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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.
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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 and temperate rain forests. These different outcomes may trace to the lack of deciduous species in tropical and temperate rain 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 effect was negative, versus positive 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