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\documentclass[a4paper,11pt]{article}
\usepackage{lmodern}
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\usepackage{amsmath}
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    \usepackage{fontspec}
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% % use microtype if available
% \IfFileExists{microtype.sty}{%
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% \usepackage{microtype}
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% \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
% }{}
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\usepackage[numbers,super,sort&compress]{natbib}
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\ifxetex
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              xetex]{hyperref}
\else
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\hypersetup{breaklinks=true,
            bookmarks=true,
            pdfauthor={Kunstler},
            pdftitle={Methods},
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\usepackage{fancyhdr}
\pagestyle{fancy}
\rhead{Methods}
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\title{Methods}
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\date{}

\begin{document}
\maketitle

\section{Model and analysis}\label{model-and-analysis}

To examine the link between competition and traits we used a
neighbourhood modelling
framework\citep{Canham-2006, Uriarte-2010, Ruger-2012, Kunstler-2012, Lasky-2014}
to model the growth of a focal tree of species \(f\) as a product of its
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maximum growth (determined by its traits and size) together with
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reductions due to competition from individuals growing in the local
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neighbourhood (see definition below). Specifically, we assumed a relationship of the form
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\begin{equation} \label{G1}
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G_{i,f,p,s,t} = G_{\textrm{max} \, f,p,s} \, D_{i,f,p,s,t}^{\gamma_f} \,  \exp\left(\sum_{c=1}^{N_i} {-\alpha_{c,f} B_{i,c,p,s}}\right),
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\end{equation}

where:

\begin{itemize}
\itemsep1pt\parskip0pt\parsep0pt
\item
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  \(G_{i,f,p,s,t}\) and \(D_{i,f,p,s,t}\) are the the annual basal area
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  growth and diameter at breast height of individual \(i\) from species
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  \(f\), plot or quadrat (see below) \(p\), data set \(s\), and census $t$,
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\item
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  \(G_{\textrm{max} \, f,p,s}\) is the maximum basal area growth for species \(f\) on plot or quadrat \(p\) in data set \(s\), i.e.~in
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  absence of competition,
\item
  \(\gamma_f\) determines the rate at which growth changes with size for
  species \(f\), modelled with a normally distributed random effect of
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  species \(\varepsilon_{\gamma, f}\) {[}as
  \(\gamma_f = \gamma_0 + \varepsilon_{\gamma, f}\) where
  \(\varepsilon_{\gamma, f} \sim \mathcal{N} (0,\sigma_{\gamma})\) -- a
  normal distribution of mean 0 and standard deviation $\sigma_{\gamma}${]}
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\item
  \(\alpha_{c,f}\) is the per unit basal area effect of individuals from
  species \(c\) on growth of an individual in species \(f\), and
\item
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  \(B_{i,c,p,s}= 0.25\, \pi \, \sum_{j \neq i} w_j \, D_{j,c,p,s,t}^2\) is
  the sum of basal area of all individuals competitor trees \(j\) of the species
  \(c\) within the local neighbourhood
  of the tree $i$ in
  plot \(p\), data
  set \(s\) and census $t$, where \(w_j\) is a constant based on
  neighboorhood size for tree $j$ depending on the data set (see
  below). Note that \(B_{i,c,p,s}\) include all trees of species $c$
  in the local neighbourhood excepted the tree
  \(i\),
\item
  \(N_i\) is the number of competitor species in the local
  neighbourhood of focal tree $i$.
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\end{itemize}

Values of \(\alpha_{c,f}> 0\) indicate competition, whereas
\(\alpha_{c,f}\) \textless{} 0 indicates facilitation.

Log-transformation of eq. \ref{G1} leads to a linearised model of the
form

\begin{equation} \label{logG1}
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\log{G_{i,f,p,s,t}} = \log{G_{\textrm{max} \, f,p,s}} + \gamma_f \, \log{D_{i,f,p,s,t}} +  \sum_{c=1}^{N_i} {-\alpha_{c,f} B_{i,c,p,s}}.
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\end{equation}

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To include the effects of traits on the parameters of the growth model we build on previous studies that explored the role of traits for tree performances and tree competition\citep{Uriarte-2010, Kunstler-2012, Lasky-2014}. We modelled the effect of trait, one trait at a time.
The effect of a focal species' trait value, \(t_f\), on its
maximum growth was include as:
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\begin{equation} \label{Gmax}
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\log{G_{\textrm{max} \, f,p,s}} = m_{0} + m_1 \, t_f + m_2 \, MAT +
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m_3 \, MAP + \varepsilon_{G_{\textrm{max}}, f} + \varepsilon_{G_{\textrm{max}}, p} + \varepsilon_{G_{\textrm{max}}, s}.
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\end{equation}

Here \(m_0\) is the average maximum growth, \(m_1\) gives the effect of
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the focal species trait, $m_2$ and $m_3$ of mean annual temperature
$MAT$ and sum of annual precipitation $MAP$ respectively, and \(\varepsilon_{G_{\textrm{max}}, f}\),
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\(\varepsilon_{G_{\textrm{max}}, p}\), \(\varepsilon_{G_{\textrm{max}}, s}\)
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are normally distributed random effect for species \(f\), plot or
quadrat \(p\) (see below), and data set \(s\) {[}where
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\(\varepsilon_{G_{\textrm{max}, f}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, f}})\);
\(\varepsilon_{G_{\textrm{max}, p}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, p}})\)
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and
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\(\varepsilon_{G_{\textrm{max}, s}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, s}})\){]}.
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Previous studies have proposed different decomposition of the competition parameter into key trait-based processes\footnote{There has been different approach to model $\alpha$ from traits. In one of the first study Uriarte et al.\citep{Uriarte-2010} modelled $\alpha$ as $\alpha =
\alpha_0 + \alpha_s \vert t_f-t_c \vert$. Then Kunstler et al.\citep{Kunstler-2012} used two different models: $\alpha = \alpha_0 + \alpha_s \vert t_f-t_c \vert$ or $\alpha =
\alpha_0 + \alpha_h ( t_f-t_c )$. Finally, Lasky et al.\citep{Lasky-2014} developped one single model inculding multiple-processes as $\alpha =
\alpha_0 + \alpha_t t_f +\alpha_h ( t_f-t_c ) + \alpha_s \vert t_f-t_c
\vert$. In this study, we extended this last model by considering that it was more clear to split
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$\alpha_h (t_f - t_c)$ in $\alpha_t t_f + \alpha_e t_c$, which is equivalent to the hierarchical distance if $\alpha_t = - \alpha_e$ (thus avoiding replication of $t_f$ effect through both $\alpha_h$ and $\alpha_t$), and splitting $\alpha_0$ in intra- and inter-specific competition.}, here we extended the approach of the most recent study\citep{Lasky-2014}. As presented in Fig. 1, competitive
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interactions were modelled using an equation of the form\footnote{For
  fitting the model the equation of \(\alpha_{c,f}\) was developped with
  species basal area in term of community weighted mean of the trait,
  see Supplementary methods for more details.}:

\begin{equation} \label{alpha}
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\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
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\end{equation}

where:

\begin{itemize}
\itemsep1pt\parskip0pt\parsep0pt
\item
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  $\alpha_{0,f,intra}$ and $\alpha_{0,f,inter}$ are respectively  intra- and interspecific trait independent competition for the focal
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  species \(f\), modelled with a normally distributed random effect of
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  species \(f\) and each with normally distributed random effect of data set
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  \(s\) {[}as
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  \(\alpha_{0,f} = \alpha_0 + \varepsilon_{\alpha_0, f}+ \varepsilon_{\alpha_0, s}\),
  where \(\varepsilon_{\alpha_0, f} \sim \mathcal{N} (0,\sigma_{\alpha_0, f})\) and
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  \(\varepsilon_{\alpha_0, s} \sim \mathcal{N} (0,\sigma_{\alpha_0, s})\){]}. And $CON$ is a binary variable taking the value one for $f=c$ (conspecific) and zero for $f \neq c$ (heterospecific),
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\item
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  \(\alpha_t\) is the \textbf{tolerance of competition} of the focal
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  species, i.e.~change in competition tolerance due to traits \(t_f\) of
  the focal tree with a normally distributed random effect of data set
  \(s\) included
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  {[}\(\varepsilon_{\alpha_t,s} \sim \mathcal{N} (0,\sigma_{\alpha_t})\){]},
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\item
  \(\alpha_{e}\) is the \textbf{competitive effect}, i.e.~change in
  competition effect due to traits \(t_c\) of the competitor tree with a
  normally distributed random effect of data set \(s\) included
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  {[}\(\varepsilon_{\alpha_i,s} \sim \mathcal{N} (0,\sigma_{\alpha_i})\){]}, and
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\item
  \(\alpha_s\) is the effect of \textbf{trait similarity}, i.e.~change
  in competition due to absolute distance between traits
  \(\vert{t_c-t_f}\vert\) with a normally distributed random effect of
  data set \(s\) included
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  {[}$\varepsilon_{\alpha_s,s} \sim \mathcal{N} (0,\sigma_{\alpha_s})${]}.
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\end{itemize}

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Estimating different $\alpha_0$ for intra- and interspecific competition allow to account for trait independant differences in interactions with conspecific or with heterospecific.
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We also explored a simpler version of the model where only one $\alpha_0$ was include in the model of $\alpha_{c,f}$ as most previous studies have generally not make this distinction which may lead into an overestimation of the trait similarity effect. In this alternative model the equation was:
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\begin{equation} \label{alpha2}
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\alpha_{c,f}= \alpha_{0,f} - \alpha_t \, t_f + \alpha_e \, t_c + \alpha_s \, \vert t_c-t_f \vert
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\end{equation}

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This results are presented in Supplementary Results.
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Eqs. \ref{logG1}-\ref{alpha} were then fitted to empirical estimates of
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growth based on change in diameter between census $t$
and $t+1$ (respectively at year $y_t$ and $y_{t+1}$), given by
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\begin{equation} \label{logGobs} G_{i,f,p,s,t} = 0.25 \pi
  \left(D_{i,f,p,s,t+1}^2 - D_{i,f,p,s,t}^2\right)/(y_{t+1} - y_t).
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\end{equation}

To estimate standardised coefficients (one type of standardised effect
size)\citep{Schielzeth-2010}, response and explanatory variables
were standardized (divided by their standard deviations) prior to
analysis. Trait and diameter were also centred to facilitate
convergence. The models were fitted using \(lmer\) in lme4\citep{Bates-2014}
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in the R statistical environment\citep{RTeam-2014}. We fitted two versions of this model. In the first
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version parameters \(m_{0}, m_1, \alpha_0,\alpha_t,\alpha_i,\alpha_s\)
were estimated as constant across all biomes. In the second version, we
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repeated the same analysis as in the first version but allowed
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different fixed estimates of these parameters for each biome. This
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enabled us to explore variation among biomes. Because some biomes had
few observations, we merged those with biomes with similar climates. Tundra was
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merged with taiga, tropical rainforest and tropical seasonal forest were
merged into tropical forest, and deserts were not included in this final
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analysis as too few plots were available. To evaluate if our results
were robust to the random effect structure we also explored a model
with a single effect constant across biomes but with a random effect
for the parameters with both the data set and a local ecoregion using
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the K{\"o}ppen-Geiger ecoregion\citep{Kriticos-2012} (see
Supplementary Results).
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\subsection{Estimating the effect of traits on the mean ratio of intra- \textit{vs.} inter-specific competition}\label{rho}
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The ratio of inter- \textit{vs.} intra-specific competition is generally considered as key in controlling species coexistence. For instance, recent studies\citep{Kraft-2015, Godoy-2014} have recently proposed to analyse the link between traits and $\rho$ defined as the geometric mean of the ratio of interspecific competition over intraspecific competition to understand traits effects on coexistence. This is based on a method developed by Chesson\citep{Chesson-2012} which demonstrates that $\rho$ can be used to quantify the stabilising niche difference between pairs of species (this estimates the strength processes favouring the establishment of a species as rare invader in the population of an other species already established, see an example based on the Lotka-Volterra model based on Godoy \& Levine\citep{Godoy-2014} in the Supplementary Methods). In this approach $\rho$ is defined as $\rho = \sqrt{\frac{\alpha'_{ij} \alpha'_{ji}}{\alpha'_{jj} \alpha'_{ii}}}$, where $\alpha'_{ij}$ represent the population level competitive effect of species $j$ on species $i$. Even if our model estimate competition effect only on the individual basal area growth, and not on the population growth, it is interesting to quantify how this ratio of inter- \textit{vs.} intra-specific competition is influenced by traits. The competitive effect of species $j$ on species $i$ can be defined in the tree basal area growth model (see equ. \ref{logG1}) as the reduction of growth of species $i$ by one unit of basal area of competitors of the species $j$ ( thus as  $\alpha'_{ij} = \frac{1}{e^{-\alpha_{ij}}}$, with $\alpha_{ij}$ defined by equ. \ref{alpha}). $\rho$ can then be related to the estimated parameters of eqn. \ref{alpha} as:
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\begin{equation} \label{rhoequ}
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\rho = \sqrt{\frac{\alpha'_{ij} \alpha'_{ji}}{\alpha'_{jj} \alpha'_{ii}}} = e^{(\alpha_{0,inter} - \alpha_{0,intra} + \alpha_s \vert t_j - t_i \vert)}
\end{equation}

The stabilising niche difference is then defined as $1-\rho$.

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\section{Data}\label{data}

\subsection{Growth data}\label{growth-data}

Our main objective was to collate data sets spanning the dominant forest
biomes of the world. Data sets were included if they (i) allowed both
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growth of individual trees and the local abundance of competitors
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to be estimated, and (ii) had good (\textgreater{}40\%) coverage for at
least one of the traits of interest (SLA, wood density, and maximum
height).

The data sets collated fell into two broad categories: (1) national
forest inventories (NFI), in which trees above a given diameter were
sampled in a network of small plots (often on a regular grid) covering
the country (references of NFI data used\citep{-b, Kooyman-2012, -e, Wiser-2001, -c, Villaescusa-1998, Villanueva-2004, Fridman-2001, -a, -d}); (2) large permanent plots (LPP) ranging in size from
0.5-50ha, in which the x-y coordinates of all trees above a given
diameter were recorded (references of LPP data used\citep{Condit-2013, Condit-1993, Lasky-2013, Ishihara-2011, Thompson-2002, Ouedraogo-2013, Herault-2010, Herault-2011} ). These LPP were mostly located in tropical
regions. The minimum diameter of recorded trees varied among sites from
1-12cm. To allow comparison between data sets, we restricted our
analysis to trees greater than 10cm. Moreover, we excluded from the
analysis any plots with harvesting during the growth measurement period,
that were identified as a plantations, or overlapping a forest edge.
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Finally, we randomly selected only two consecutive census dates per
plot or quadrat to
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avoid having to account for repeated measurements, as less than a third
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of the data had repeated measurements. Because human and natural disturbances are present in all these forests (see Supplementary Methods), they probably all experience successional dynamics (as shown by the forest age distribution available in some of these sites in Supplementary Methods). See the Supplementary Methods and
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Extended Data Table 1 for more details on the individual data sets.

Basal area growth was estimated from diameter measurements recorded
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between the two census. For the French NFI, these data were
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obtained from short tree cores. For all other data sets, diameter at
breast height (\(D\)) of each individual was recorded at multiple census
dates. We excluded trees (i) with extreme positive or negative diameter
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growth measurements, following criteria developed at the BCI site
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\citep{Condit-1993} (see the R package
\href{http://ctfs.arnarb.harvard.edu/Public/CTFSRPackage/}{CTFS R}),
(ii) that were a palm or a tree fern species, or (iii) that were
measured at different height in two consecutive censuses.

For each individual tree, we estimated the local abundance of competitor
species as the sum of basal area for all individuals \textgreater{} 10cm
diameter within a specified neighbourhood. For LPPs, we defined the
neighbourhood as being a circle with 15m radius. This value was selected
based on previous studies showing the maximum radius of interaction to
lie in the range
10-20m\citep{Uriarte-2004, Uriarte-2010}. To avoid
edge effects, we also excluded trees less than 15m from the edge of a
plot. To account for variation of abiotic conditions within the LPPs, we
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divided plots into regularly spaced 20x20m quadrats and included in a
random quadrat effect (see above).
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For NFI data coordinates of individual trees within plots were generally
not available, thus neighbourhoods were defined based on plot size. In
the NFI from the United States, four sub-plots of 7.35m located within
20m of one another were measured. We grouped these sub-plots to give a
single estimate of the local competitor abundance. Thus, the
neighbourhoods used in the competition analysis ranged in size from
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10-25 m radius, with most plots 10-15 m radius. We included variation in
neighbourhood size in the constant $w_j$ to compute competitors basal
area in $m^2/ha$.
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We extracted mean annual temperature (MAT) and mean annual sum of
precipitation (MAP) from the \href{http://www.worldclim.org/}{worldclim}
data base \citep{Hijmans-2005}, using the plot latitude and
longitude. MAT and MAP data were then used to classify plots into
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biomes, using the diagram provided by Ricklefs\citep{Ricklefs-2001}
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(after Whittaker).

\subsection{Traits}\label{traits}

Data on species functional traits were extracted from existing sources.
We focused on wood density, species specific leaf area (SLA) and maximum
height, because these traits have previously been related to competitive
interactions and are available for large numbers of species
\citep{Wright-2010, Uriarte-2010, Ruger-2012, Kunstler-2012, Lasky-2014}
(see Extended data Table 2 for traits coverage). Where available we used
data collected locally (References for the local traits data used in this analysis\citep{Wright-2010, Swenson-2012, Gourlet-Fleury-2011, Lasky-2013, Baraloto-2010}); otherwise we sourced data from the
\href{http://www.try-db.org/}{TRY} trait data base
\citep{Kattge-2011} (References for the data extracted from the TRY database used in this analysis\citep{Ackerly-2007, Castro-Diez-1998, Chave-2009, Cornelissen-1996, Cornelissen-1996a, Cornelissen-1997, Cornelissen-2004, Cornelissen-2003, Cornwell-2009, Cornwell-2006, Cornwell-2007, Cornwell-2008, Diaz-2004, Fonseca-2000, Fortunel-2009, Freschet-2010, Freschet-2010a, Garnier-2007, Green-2009, Han-2005, He-2006, He-2008, Hoof-2008, Kattge-2009, Kleyer-2008, Kurokawa-2008, Laughlin-2010, Martin-2007, McDonald-2003, Medlyn-1999a, Medlyn-1999, Medlyn-2001, Messier-2010, Moles-2005b, Moles-2005a, Moles-2004, Niinemets-2001, Niinemets-1999, Ogaya-2003, Ogaya-2006, Ogaya-2007, Ogaya-2007a, Onoda-2011, Ordonez-2010, Ordonez-2010a, Pakeman-2008, Pakeman-2009, Penuelas-2010, Penuelas-2010a, Poorter-2006, Poorter-2009, Poorter-2009a, Preston-2006, Pyankov-1999, Quested-2003, Reich-2008, Reich-2009, Sack-2004, Sack-2005, Sack-2006, Sardans-2008, Sardans-2008a, Shipley-2002, Soudzilovskaia-2013, Willis-2010, Wilson-2000, Wright-2007, Wright-2006, Wright-2010, Wright-2004, Zanne-2010}).
 Local data were available for most tropical
sites and species (see Supplementary methods). Several of the NFI data
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sets also provided tree height measurements, from which we computed a
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species' maximum height as the 99\% quantile of observed values (for
France, US, Spain, Switzerland). For Sweden we used the estimate from
the French data set and for Canada we used the estimate from the US data
set. Otherwise, we extracted measurement from the TRY database. We were
not able to account for trait variability within species between sites.

For each focal tree, our approach required us to also account for the
traits of all competitors present in the neighbourhood. Most of our
plots had good coverage of competitors, but inevitably there were some
trees where trait data were lacking. In these cases we estimated trait
data as follows. If possible, we used the genus mean, and if no genus
data was available, we used the mean of the species present in the
country. However, we restricted our analysis to plots where (i) the
percentage of basal area of trees with no species level trait data was
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less than 10\%, and (ii) the
percentage of basal area of trees with no species and genus level trait data was
less than 5\%.
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\newpage
\clearpage
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\section{References}\label{references}

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\bibliographystyle{naturemag}
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\bibliography{references}

\end{document}