# Calculating functional diversity metrics using neighbor-joining trees

Pedro Cardoso, Thomas Guillerme, Stefano Mammola, Thomas Matthews, François Rigal, Caio Graco-Roza, Gunilla Stahls, Jose Carlos Carvalho

► To cite this version:

Pedro Cardoso, Thomas Guillerme, Stefano Mammola, Thomas Matthews, François Rigal, et al. Calculating functional diversity metrics using neighbor-joining trees. Ecography, 2024, 2024 (7), pp.e07156. 10.1111/ecog.07156. hal-04935128

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ECOGRAPHY

Research article

Calculating functional diversity metrics using
neighbor-joining trees

Pedro Cardoso<sup>1,2</sup>, Thomas Guillerme<sup>3</sup>, Stefano Mammola<sup>2,4</sup>, Thomas J. Matthews<sup>5,6</sup>, Francois Rigal<sup>6,7</sup>, Caio Graco-Roza<sup>8,9</sup>, Gunilla Stahls<sup>2</sup> and Jose Carlos Carvalho<sup>6,10</sup>

**Correspondence:** Pedro Cardoso (pedro.cardoso@helsinki.fi)

Ecography

2024: e07156

doi: 10.1111/ecog.07156

Subject Editor: Timothée Poisot

Editor-in-Chief: Miguel Araújo

Accepted 18 March 2024

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

The study of functional diversity (FD) provides ways to understand phenomena as com-
plex as community assembly or the dynamics of biodiversity change under multiple
pressures. Different frameworks are used to quantify FD, either based on dissimilarity
matrices (e.g. Rao entropy, functional dendrograms) or multidimensional spaces (e.g.
convex hulls, kernel-density hypervolumes), each with their own strengths and limits.
Frameworks based on dissimilarity matrices either do not enable the measurement of all
components of FD (i.e. richness, divergence, and regularity), or result in the distortion
of the functional space. Frameworks based on multidimensional spaces do not allow for
comparisons with phylogenetic diversity (PD) measures and can be sensitive to outliers.
We propose the use of neighbor-joining trees (NJ) to represent and quantify FD in a
way that combines the strengths of current frameworks without many of their weak-
nesses. Importantly, our approach is uniquely suited for studies that compare FD with
PD, as both share the use of trees (NJ or others) and the same mathematical principles.
We test the ability of this novel framework to represent the initial functional distances
between species with minimal functional space distortion and sensitivity to outliers.
The results using NJ are compared with conventional functional dendrograms, convex
hulls, and kernel-density hypervolumes using both simulated and empirical datasets.
Using NJ, we demonstrate that it is possible to combine much of the flexibility provided
by multidimensional spaces with the simplicity of tree-based representations. Moreover,
the method is directly comparable with taxonomic diversity (TD) and PD measures, and
enables quantification of the richness, divergence and regularity of the functional space.

<footnote n="1">¹CE3C - Centre for Ecology, Evolution and Environmental Changes, CHANGE - Global Change and Sustainability Institute, Faculty of Sciences, University of Lisbon, Portugal</footnote>

<footnote n="2">²Laboratory for Integrative Biodiversity Research, Finnish Museum of Natural History Luomus, University of Helsinki, Helsinki, Finland</footnote>

<footnote n="3">³Department of Animal and Plant Sciences, The University of Sheffield, Sheffield, UK</footnote>

<footnote n="4">⁴Molecular Ecology Group (MEG), Water Research Institute, National Research Council (CNR-IRSA), Verbania Pallanza, Italy</footnote>

<footnote n="5">⁵GEES (School of Geography, Earth and Environmental Sciences) and Birmingham Institute of Forest Research, University of Birmingham, Birmingham, UK</footnote>

<footnote n="6">⁶CE3C - Centre for Ecology, Evolution and Environmental Changes, CHANGE - Global Change and Sustainability Institute, Faculty of Sciences, University of the Azores, Angra do Heroismo, Portugal</footnote>

<footnote n="7">⁷CNRS - Université de Pau et des Pays de l'Adour - E2S UPPA, Institut des Sciences Analytiques et de Physico Chimie pour L'environnement et les Materiaux, UMR5254, Pau, France</footnote>

<footnote n="8">⁸Lammi Biological Station, University of Helsinki, Lammi, Finland</footnote>

<footnote n="9">⁹Department of Geosciences and Geography, University of Helsinki, Finland</footnote>

<footnote n="10">¹⁰Molecular and Environmental Centre - Centre of Molecular and Environmental Biology, Department of Biology, University of Minho, Braga, Portugal</footnote>
Introduction

With the advent of new ways of thinking about biodiversity (McGill et al. 2006) and novel sources of data (Jarić et al. 2020, Tosa et al. 2021, Tobias et al. 2022), we are experiencing a shift from measuring biodiversity based on species identities only (taxonomic diversity, TD), to taking into account either the evolutionary relationships among species (phylogenetic diversity, PD) or similarities in functional traits (functional diversity, FD) (Pavoine and Bonsall 2011). An integrative approach to quantifying biodiversity enables not only the comparison of its multiple facets (Pollock et al. 2020), but provides new tools to understand phenomena as complex as community assembly and the dynamics of biodiversity change under multiple pressures (McGill et al. 2006).

Both PD and FD can be measured within (alpha diversity) and between (beta diversity) samples, sites, or time steps (Whittaker 1960). Diversity can also be measured in terms of richness, divergence and regularity (Pavoine and Bonsall 2011, Tucker et al. 2017, Mammola et al. 2021). For PD, these facets are usually quantified using dissimilarity matrices, calculated either directly from the raw dissimilarity (e.g. Rao entropy, Botta-Dukát 2005, Hill numbers, Chao et al. 2014) or from phylogenetic trees (Tucker et al. 2017). For FD, multidimensional approaches reflecting the niche concept by Hutchinson (1957) are often used, with multiple advantages (Blonder 2016, Carvalho and Cardoso 2020, Mammola and Cardoso 2020, Mammola et al. 2021) over tree-based metrics (Petchey and Gaston 2002, 2006). For example, functional trees are usually built using hierarchical clustering methods such as the unweighted pair group method using arithmetic averages (UPGMA; Michener and Sokal 1957, Cardoso et al. 2014) or single-linkage trees (equivalent to minimum spanning trees; Villéger et al. 2008), and it has been shown that, in comparison, hyperdimensional approaches better maintain the original distances between species, which eliminates or at least minimises the distortion of the functional space (Maire et al. 2015).

Trees and hyperdimensional representations however require the use of different methods with non-comparable mathematical properties (Mammola et al. 2021). As such, when comparing phylogenetic and functional diversity, it is impossible to know if any differences observed in index values between samples (e.g. richness) are due to inherent differences of the studied systems or due to the use of different algorithms. Using dissimilarity matrices or trees is the only option to compare PD and FD. As many studies use phylogenetic trees, and many regularity and beta diversity partitioning metrics are exclusively calculated using functional tree representations, the use of trees is often preferred for PD/FD comparisons (Cardoso et al. 2014).

Phylogenetic tree reconstruction has seen major advances in recent decades due to the development of ever more

efficient algorithms for the representation of evolutionary relationships (Nguyen et al. 2015, Minh et al. 2020). Once among the most used, the neighbor-joining (NJ) method reconstructs (phylogenetic) trees from evolutionary distance data (Saitou and Nei 1987). The algorithm for constructing NJ trees connects the terminals based on their overall similarity, and continues to be widely used as it is known to be both efficient and computationally fast. It has a much better performance for reconstructing distance-based trees than UPGMA (Saitou and Nei 1987). Even if other methods can outperform it under different evolutionary scenarios (Rannala and Yang 1996), NJ trees are still widely used for preliminary similarity clustering at the species level. As an example, the Taxon ID tree tool in BOLD (Ratnasingham and Hebert 2007) employs NJ under the Kimura two-parameter (K2P; Kimura 1980) distance metric. At the species level, NJ performs best when the mutational rate heterogeneity among the terminals is low, and typically resolves a well-sampled input distance matrix consistently (Atteson 1997). The tree topology is also correct if each entry in the distance matrix differs from the true distance by less than half of the shortest branch length in the tree (Mihaescu et al. 2009). As both NJ and UPGMA lack an optimality criterion defining the best tree, an analysis returns only one optimal topology.

All diversity measures calculated using phylogenetic trees can also be calculated using functional trees built with the neighbor-joining algorithm with no changes required, this way allowing comparison of, for example, phylogenetic trees constructed using Bayesian methods with NJ trees built for functional diversity. The construction of functional trees is based on pairwise trait dissimilarities between species, a roughly analogous approach to using genetic dissimilarities to build phylogenetic trees. The comparison between phylogenetic and functional diversity can be made as the algorithms used to calculate them are the same, even if the trees were built using different methods and have different meaning.

In this work, we propose and test NJ trees as a way to quantify functional richness, divergence and regularity without the functional space distortion typical of functional dendrograms built using hierarchical clustering. We provide functions for all methods in the R package 'BAT' (www.r-project.org, Cardoso et al. 2015).

Material and methods

Building NJ trees requires a distance matrix and in the construction of phylogenetic trees the principle of parsimony is used. The algorithm builds a non-ultrametric tree, in a way that the branching patterns and branch lengths are optimised to minimise the amount of change needed to connect all species along the tree (Saitou and Nei 1987). Hence, the
<figure><img src="image_5.png" /><figcaption>Figure 1. Top-left: NJ tree representing evolutionary or functional distances (edges, in black) connecting four hypothetical species A to D (nodes, in blue) with the root set in species B. A:B=1; A:C=A:D=7; B:C=B:D=6; C:D=4. Top-centre and top-right: the same tree for two communities with species A, B and C, and A, B and D respectively. In the middle we represent the calculation of different metrics for community 1. At the bottom, different metrics for species A within community 1 (main text).</figcaption></figure>

total length of the tree, equivalent to a measure of phyloge-
netic richness, is also minimised (Fig. 1). Here we propose
to use the same principle to build functional trees depict-
ing distances between species (function *BAT::tree.build*). For
taxonomic diversity, a star-like tree could be used, with all
pairwise distances being equal to one, this way guaranteeing
comparability for TD, PD and FD.

In the context of FD, the same algorithm is used but
replacing phylogenetic distance with the functional distance
between the species. Similar to the concept of hierarchical
clustering or minimum spanning trees (which are in effect
equivalent to hierarchical clustering with single linkage;
Gower and Ross 1969), NJ trees are much more flexible
due to two characteristics. First, they do not explicitly build
ultrametric trees as in UPGMA and related methods. Second,
while in traditional methods species are directly assigned to
nodes, NJ trees use intermediate nodes which are not directly
assigned to species. Such flexibility results in cophenetic dis-
tances much closer to the initial distances, decreasing the
known limitations of hierarchical clustering that distorts this
relation (Maire et al. 2015).

When calculating FD, the distances between species can be
Euclidean if all traits are continuous, and traits should gener-
ally be standardised (e.g. z-score) to ensure the weight of each is
similar (BAT::standard). As categorical or ordinal traits are com-
monly available, Gower's distance is often used (Pavoine et al.
2009, BAT::gower), and it is possible to weight traits if some are
considered more important with regard to how species interact
with their environment, including other species. As traits are
often correlated, one might want to first perform a principal
coordinates analysis (PCoA) or similar technique and use the
resulting orthogonal axes as the new traits.

For all PD/FD analyses one should always use the same
phylogenetic or functional tree depicting the relationships

between all species, to guarantee the comparability of results.
Phylogenetic or functional richness is the sum of lengths of
edges connecting all species in a community (Faith 1992,
Petchey and Gaston 2002, 2006, BAT::alpha). For the com-
munities in Fig. 1, functional richness would be Community
1=Community 2=7. Species-level measures can be calcu-
lated in different ways. Originality of a species is measured
as its average distance to all other species in a community
(Pavoine et al. 2005, BAT::originality). In the example, for
Community 1 the originality would be A=4, B=3.5 and
C=6.5. Uniqueness of a species is measured as its distance
to the single closest species in the community (Mouillot et al.
2013, BAT::uniqueness). In Community 1 it would be A=1,
B=1, and C=6. The contribution of a species to richness
or alpha diversity is the length of branches unique to it, plus
the proportional length of shared branches connecting it to
the root of the tree (Isaac et al. 2007, BAT::contribution).
As NJ trees are unrooted, they can either be rooted using an
outgroup or in the midpoint (Podani et al. 2000). Using an
outgroup can be justified in certain cases, for example if one
species is known to represent the ancestral state from which
all others have evolved a set of traits. In most cases however
midpoint rooting is a better option. In this alternative the two
species with the highest pairwise distance are selected, and
the root is placed halfway between them (Podani et al. 2000).
In Community 1, contribution would be A=1, B=0, and
C=6. This option for rooting is however mostly arbitrary,
and alternatives could be explored in the future.

The second dimension of phylogenetic or functional
alpha diversity is divergence (Mammola et al. 2021). It can
be calculated as the average dissimilarity between any two
species or individuals randomly chosen in a community
(BAT::dispersion). If abundance data are used, dispersion is
the quadratic entropy of Rao (1982), otherwise it is the phy-
logenetic dispersion measure of Webb et al. (2002), often
referred to in the phylogenetic diversity literature as mean
pairwise distance (MPD). In the example of Fig. 1, if all
species abundances are 1, dispersion would be Community
1=Community 2=4.667, i.e. the average value of all three
pairwise distances (A:B=1, B:C=6, A:C=7). One might
also calculate dispersion using the average distance to the
closest species, usually termed mean nearest taxon distance
(MNTD).

The third dimension of phylogenetic or functional alpha diversity is regularity (Mammola et al. 2021). It represents the evenness in the abundances and distances between connected species in a community (BAT::evenness). It can be calculated, among other approaches, using the index of Camargo (1993), which we refer to for an explanation of the index. In the example of Fig. 1, if all species abundances are 1, evenness would be Community 1 = Community 2 = 0.754.

Finally, beta diversity represents the dissimilarity between
two communities (BAT::beta; measured using either
Jaccard or Sørensen dissimilarity) and can be partitioned
into the two processes contributing to it, replacement and
loss or gain of species leading to differences in richness
(Carvalho et al. 2012), evolutionary history, or functional
traits (Cardoso et al. 2014). In the example, comparing communities 1 and 2, $\beta_{total} = \beta_{repl} = 0.444$, and $\beta_{rich} = 0$.

## Comparing frameworks using simulated scenarios

We simulated trees using a birth-death model, with both birth (speciation) and death (extinction) parameters drawn from a uniform distribution (0, 1) while keeping the death parameter lower than the birth parameter. For each lineage simulated in the birth-death process, we also simultaneously simulated a trait value as a function of branch length. The traits were simulated using either: 1) a Brownian motion process (BM; whereby the trait value at time $t+1$ is independent of its value at time $t$, resulting in an increase in trait variance through time); or 2) an Ornstein-Uhlenbeck process (OU; whereby the trait value at time $t+1$ is independent of its value at time $t$ but constrained by an overall parameter, alpha, that effectively reduces the increase in trait variance through time). For each tree simulation, we chose the trait process randomly between both processes described above. We ran the birth-death and trait simulations until reaching 100 co-occurring species. For each simulation, we then discarded the extinct species resulting in trees with 100 tips with 1 trait value each. We ran the birth-death and trait simulations using the R package 'treats' (Guillerme 2023).

For each of the two evolutionary processes, we simulated 1, 2, 4, or 8 orthogonal (i.e. uncorrelated) traits, 10 runs per trait number combination, totalling 40 runs per process. For each of these 40 runs the simulation created a functional tree for 100 extant species. For each run, we then sampled 10 communities with an increasing number of species (10, 20, ..., 100 species), reaching a final sample size of 800 (2 processes * 4 sets of number of traits * 10 runs * 10 communities).

For each community, we first estimated the correlation between functional richness, divergence, and regularity calculated with NJ trees, checking whether the three metrics were able to capture distinct facets of FD (which is achieved when correlation is low; Mouchet et al. 2010). Next, we used the 'BAT' R package to estimate and compare functional richness (functions *alpha*, *hull.alpha* and *kernel.alpha*), divergence (functions *dispersion* and *kernel.dispersion*), and regularity (functions *evenness* and *kernel.evenness*) with NJ trees, UPGMA trees, convex hulls, and kernel-density n-dimensional hypervolumes using the algorithms described above for NJ and UPGMA trees and those described in Carvalho and Cardoso (2020) and Mammola and Cardoso (2020) for hypervolumes. Convex hulls were only used to estimate richness, as divergence and regularity are not possible to calculate with such a representation (Mammola et al. 2021). We used Spearman's correlation coefficient to assess the correlation among the estimates of functional richness, divergence, and regularity obtained by the different frameworks.

## Functional space quality and sensitivity to outliers

When building a functional space, a crucial aspect is to assess its quality, i.e. the extent to which the functional space is an

accurate representation of the initial trait values. In order to achieve this goal, for each pair of species i and j, we compared the initial dissimilarity distance ($d_{ij}$) with the distance in the functional space ($h_{ij}$) obtained by NJ, UPGMA and PCoA (multidimensional space) methods. For the NJ and UPGMA trees, $h_{ij}$ corresponded to the cophenetic distance between species i and j. For the PCoA, we calculated the Euclidean distance between the coordinates of species i and j in the space defined by the PCoA axes. We then calculated the quality of the representation of the functional spaces using the same three frameworks (NJ, UPGMA and PCoA) using the functions *BAT::tree.quality* and *BAT::hyper.quality* (the latter being used for any representation using hyperspaces, i.e. convex hulls or kernel-density hypervolumes). Both these functions calculate the inverse of mean squared deviation between initial and cophenetic distances (Maire et al. 2015) after standardisation of all values between 0 and 1 for simplicity of interpretation and comparability of trees and multidimensional spaces.

The quality of the functional spaces was evaluated in 10 simulations for each combination of number of species per community (from 10 to 100 species), number of traits per species (one, two, four and eight) and evolutionary processes used to generate the traits (BM and OU). For PCoA we did not assess the quality for single traits. It is worth noting that the maximum number of PCoA axes that can be extracted from a matrix of N continuous traits is N. Hence, the quality of the functional space is 1 when using N axes. Therefore, we only used the simulated datasets with eight traits to assess the quality of the functional spaces built by PCoA.

We used linear mixed models to estimate the effect of the different methods (fixed effects) on the quality of the functional space. The number of species per community, the number of traits per species and the evolutionary process used to generate the traits were introduced in the models as random effects. Because the quality of the functional space ranges between 0 and 1, with true 1s but no true 0s included in the response, we transformed all 1s by subtracting 0.0001 and then ran the models using a beta distribution with a logit link function. Mixed models were performed using the 'glmmTMB' (Mollie et al. 2024) package, and model validation was performed by checking heterocedasticity, posterior predictive checking, and normality of random effects and residuals using the 'performance' (Lüdecke et al. 2021) package in R.

Finally, the sensitivity to outliers was also compared by deleting the species with higher uniqueness in each community and calculating the percentage of change in the values of richness before and after deletion. For the multidimensional space, we calculated differences in richness for kernel-density hypervolumes.

## Comparing frameworks using empirical data

The study of avian functional diversity has recently gained momentum due to the release of the AVONET database, which provides a complete set of data for eight continuous morphological traits for all the world's extant bird species
(Tobias et al. 2022). Dozens of papers have been published using this data source in just a few years (Weeks et al. 2022), including several focusing on islands (Matthews et al. 2022, Soares et al. 2022). An often-mentioned issue when studying the functional diversity of birds is the so-called 'kiwi problem'. In short, kiwis (Apterygidae: *Apteryx* spp.) differ substantially from other birds regarding their morphology (e.g. for wing length, in AVONET, the kiwis have arbitrary values that are roughly 267 times smaller than the species with the next smallest wing length) and thus all five species are always (extreme) outliers in functional diversity analyses. As such, there can be large differences in (functional) richness depending on whether they are included when building the functional space or not (Matthews et al. 2022, see also Fig. 1 in Pigot et al. 2020). These differences reflect the sensitivity to outliers of multidimensional representations such as convex hulls and kernel-density hypervolumes and researchers often opt not to include these species (Stewart et al. 2023).

To test the sensitivity to outliers of UPGMA, NJ, convex hulls and kernel-density n-dimensional hypervolumes, we took the five kiwi species and then randomly selected 100 bird species from the global species pool, making sure to include one representative from each order. For these 105 species, we sourced data on eight continuous traits (total beak length from the tip to the skull, beak length to the nares, beak width and depth at the nares, wing length, secondary length, tail length, and tarsus length) from AVONET (Tobias et al. 2022). Traits were log-transformed and scaled to mean=0, SD=1. We then undertook a PCoA and took the first five axes that summed to 99% explained variance to avoid the use of correlated variables. We used these five axes to build the convex hulls, kernel hypervolumes and UPGMA and NJ trees; for the latter, Euclidean distances between species were first calculated. Functional richness was calculated with the *BAT::alpha*, *BAT::hull.alpha* and *BAT::kernel.alpha* functions. We then re-calculated functional richness after removing the kiwis from the community and quantified the percent loss. In addition, we quantified the tree and hyperspace qualities as above.

## Results

In regard to the simulations, the correlation between richness, divergence and regularity was low for all trait combinations, except for richness versus regularity which attained values around 0.7 or above (Fig. 2). Richness and regularity were also sensitive to the number of species ($r > 0.6$ for all trait combinations), as expected, at least for richness. We found a very high convergence among the estimations based on NJ and UPGMA trees, irrespective of the number of traits and the facets of FD (Fig. 2). Correlations were lower between NJ and hypervolumes, especially for the divergence and regularity components, but also for the richness component in high dimensions (eight traits).

The quality of the simulated functional spaces obtained by the NJ method was superior to those constructed by

UPGMA, for all the combinations of number of species per community (from 10 to 100 species), number of traits per species (one, two, four and eight), and evolutionary processes used to generate the traits (BM and OU) (Supporting information). It is worth mentioning that the quality of the functional spaces constructed with NJ for communities with only one trait is always 1. Likewise, the quality of the functional space for indices using raw distances, such as Rao's entropy is, by definition, always 1. Results of the mixed model analysis confirmed that NJ performance, in terms of functional space quality, was significantly better than UPGMA (Table 1).

The quality of the functional spaces built using the NJ method was similar to that obtained with four dimensions (half the maximum number of axes) using PCoA (Table 2). The performance of NJ was higher than multidimensional spaces with two or three dimensions, but lower than multidimensional spaces with more than four dimensions (Table 2). One should note that the results for multidimensional spaces can be applied to both convex hulls and kernel-density hypervolumes, as quality is measured on the space itself, which is common to both approaches. A degree of caution is required when interpreting the mixed model results. Specifically, the variance estimates associated with the evolutionary processes (variable with 2 levels) and the number of traits (variable with 4 levels) should be regarded cautiously, due to the small number of levels involved. Nevertheless, it must be emphasised that the main purpose of fitting the model was to compare the performance of methods in terms of functional space quality and not the effects of evolutionary processes and the number of traits per se.

Regarding the sensitivity to outliers, NJ and UPGMA were found to be similar for all simulations regardless of the number of traits (Supporting information). Other indices such as Rao's entropy also use dissimilarity matrices and hence their performance is similar to any kind of tree. In contrast, hypervolumes were more sensitive, with higher differences between initial and final richness values after excluding the most unique species in each community for most scenarios.

In regard to the empirical test focused on the 'kiwi problem', the quality of the functional space for birds using NJ was 0.994, compared with 0.953 for UPGMA and 0.996 for two PCoA axes (> 0.999 for three or more axes). The exclusion of kiwis from the community, i.e. decreasing ~ 5% of the species richness, led to a decrease in functional richness of 10 and 14% for UPGMA and NJ respectively. In contrast, functional richness as measured using convex hulls was reduced by 73% and using kernel density hypervolumes by 42% (Fig. 3).

## Discussion

The study of functional diversity is a burgeoning research area in ecology and evolution, with numerous methodological developments during the last couple of decades (De Bello et al. 2021, Mammola et al. 2021, Palacio et al. 2022). In contrast to the study of taxonomic or phylogenetic
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 2. Pairwise Spearman's correlations among estimations of functional richness, divergence, and regularity based on neighbor-joining (NJ) trees (A), and among estimations of functional richness (B), divergence (C), and regularity (D) with NJ trees, unweighted pair group method using arithmetic averages (UPGMA) trees, and kernel-density n-dimensional hypervolumes. Only Brownian motion process simulations are shown for simplicity, Ornstein-Uhlenbeck simulations show similar patterns. For each panel plot, density plots on the diagonal display the distribution of values. Bivariate scatter plots are displayed below the diagonal and the correlation values above the diagonal. For tree representations, we used the framework as described in Fig. 1. For hypervolumes, we used the algorithms described in Carvalho and Cardoso (2020) and Mammola and Cardoso (2020). Results for convex hulls are not presented as divergence and regularity are not possible to calculate (Mammola et al. 2021).</figcaption></figure>

diversity, where the methodological approaches to quantify
diversity are relatively established, there is still much discus-
sion around how to best represent and measure FD across
its dimensions, namely richness, divergence and regularity.
Raw data, different tree representations, or representations
based on multidimensional spaces, all have their strengths
and weaknesses (Mammola et al. 2021). Here we propose a
novel approach to measuring FD using trees constructed with
the NJ algorithm that combines many of the advantages of
these different approaches, while mitigating the limitations.

We stress that the same approach can be applied to
trees constructed with any algorithm, from hierarchical

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

clustering to maximum parsimony or maximum likelihood,
thus enabling straightforward comparisons of FD with PD.

Statistical properties

Our results indicate that the NJ method is more accurate
than UPGMA (and similar methods such as minimum span-
ning trees) in representing the functional space occupied by
a given community, i.e. the quality of the trait space. It is on
par with multidimensional representations with up to four
dimensions in simulated scenarios that cover a large variety
of real-world situations. Given the ‘curse of dimensionality’
<table><caption>Table 1. Summary of a mixed model for the quality of functional space, where we included method [neighbor-joining (NJ) and unweighted pair group method using arithmetic averages (UPGMA)] as a fixed parameter predictor (fixed) and allowed the intercept to vary (random) across number of traits per species within blocks of number of species per community, and across evolutionary processes. Significant estimates are in bold.</caption><thead><tr><th>Effect</th><th>Predictor</th><th>Estimate</th><th>Std. error</th><th>Statistic</th></tr></thead><tbody><tr><td rowspan="2">Fixed</td><td>(Intercept)</td><td>3.687</td><td>0.413</td><td><strong>8.933</strong></td></tr><tr><td>UPGMA</td><td>-2.393</td><td>0.043</td><td><strong>-55.257</strong></td></tr><tr><td>Random</td><td>sd(Intercept)</td><td>0.523</td><td></td><td></td></tr><tr><td>Evolutionary processes</td><td>sd(Intercept)</td><td>0</td><td></td><td></td></tr><tr><td>Number of traits:</td><td>sd(Intercept)</td><td>0.357</td><td></td><td></td></tr><tr><td>Number of species</td><td>sd(Intercept)</td><td>0.357</td><td></td><td></td></tr></tbody></table>

(Bellman 1957) of the hyperspace – the mathematical and computational difficulty of dealing with many dimensions simultaneously, and the implicit negative relationship between the number of dimensions and the volume of the hyperspace – a decrease in correlation between NJ trees and hypervolumes with increasing number of dimensions is expected. In general, NJ will be as accurate as hypervolumes in many situations and will present only small differences otherwise. To be fair, we are not arguing that NJ trees necessarily represent better quality functional spaces than multidimensional representations (at least those with four or more axes). Rather, our assertion is that, in many cases, it is desirable to calculate FD using trees (e.g. to enable easier comparison with PD, discussed below), and we have shown here that the NJ method produces higher quality trees than the commonly used UPGMA method.

<table><caption>Table 2. Summary of the mixed model for the quality of functional space, where we included method [neighbor-joining tree and from two to eight dimensions corresponding to the axes provided by a principal coordinates analysis (PCoA)] as a fixed parameter predictor (fixed) and allowed the intercept to vary (random) across the number of species per community and evolutionary processes. Significant estimates in bold.</caption><thead><tr><th>Effect</th><th>Predictor</th><th>Estimate</th><th>SE</th><th>Statistic</th></tr></thead><tbody><tr><td rowspan="7">Fixed</td><td>(Intercept)</td><td>3.467</td><td>0.235</td><td><strong>14.739</strong></td></tr><tr><td>PCoA-2 axes</td><td>-1.495</td><td>0.042</td><td><strong>-35.941</strong></td></tr><tr><td>PCoA-3 axes</td><td>-0.618</td><td>0.046</td><td><strong>-13.436</strong></td></tr><tr><td>PCoA-4 axes</td><td>0.163</td><td>0.053</td><td>3.067</td></tr><tr><td>PCoA-5 axes</td><td>0.918</td><td>0.063</td><td><strong>14.652</strong></td></tr><tr><td>PCoA-6 axes</td><td>1.607</td><td>0.072</td><td><strong>22.298</strong></td></tr><tr><td>PCoA-7 axes</td><td>1.607</td><td>0.072</td><td><strong>22.298</strong></td></tr><tr><td>PCoA-8 axes</td><td>2.729</td><td>0.082</td><td><strong>33.253</strong></td></tr><tr><td>Random</td><td>sd(Intercept)</td><td>0.121</td><td></td><td></td></tr><tr><td>Number of species</td><td>sd(Intercept)</td><td>0.324</td><td></td><td></td></tr><tr><td>Evolutionary process</td><td>sd(Intercept)</td><td>0.324</td><td></td><td></td></tr></tbody></table>

Multidimensional representations are known to have difficulties dealing with outliers, with substantially unique observations having disproportionate effects on the quantification of FD. In the empirical example illustrating the ‘kiwi problem’ (Fig. 3), excluding the kiwis from the hypervolume construction does not just result in the loss of the space they occupy, but the space representing the remaining 100 species also ‘shrinks’, as the average distance between species decreases (using consistent bandwidth values when constructing the hypervolumes may mitigate this issue, at least for Gaussian kernel density hypervolumes). For the very commonly used convex hulls, the loss is even more severe, as it includes all the intervening functional space (i.e. the space between the kiwis and all other birds where the convex hull extends out) that is in fact not occupied by any existing bird species. NJ trees can circumvent the ‘kiwi problem’ by generating a representation that is less sensitive to the large functional differences between kiwis and the remaining birds, but that is of higher quality than UPGMA trees.

## Comparing different facets of diversity

As with UPGMA and other tree methods, taxonomic diversity can be represented as a star-like NJ tree, and in fact the construction of a NJ tree starts with a starlike tree. This means that TD and FD are comparable using the same methods, although for TD they are usually simplified for speed and ease of use.

Crucially, we demonstrate that hyperspatial representations are not comparable with tree representations that are often used for quantifying PD. As seen in Fig. 2, even for the same data, tree and hypervolume values of richness, divergence or regularity have little to no correlation. This implies that, if one uses phylogenetic trees to measure PD and hypervolumes to measure FD, any differences in patterns will be due to both differences in community composition and the mathematical properties of the indices, with no possibility to disentangle these two effects. We should note that trees used for quantifying PD can be built using numerous methods, including NJ, Bayesian or any other that results in a tree (ultrametric or not, dated or undated). Even if the interpretation of the trees is necessarily different, the mathematics used to calculate richness, divergence or regularity will be similar and hence comparability is warranted.

## Advantages of neighbor-joining trees

As with other tree representations, NJ works directly with distances between species. The choice of distance is thus critical, although Gower's distance is often preferred as it accommodates different types of variables, such as continuous, ordinal, binary and categorical traits (Pavoine et al. 2009). When only continuous variables are used, as in our empirical example, Euclidean distances are generally preferred. In any case, this decision is almost always simpler to make than the ones involved in the use of multidimensional methods, which include the number of axes to use, which method to use for
<figure><img src="image_10.png" /><figcaption>Figure 3. NJ functional tree with the edge leading to the five kiwi species highlighted by the silhouette, and the multidimensional space of studied birds with (grey) and without (blue) kiwis included. Five axes were selected for analyses, adding to 99% of explained variance. Note that, when excluding the kiwis, it is not just the space they occupy that is lost, but the space representing the remaining 100 species also 'shrinks', as the average distance between species decreases. For convex hulls the difference is even higher, as all the empty intervening space is also lost. Kiwi silhouette by Ferran Sayol.</figcaption></figure>

estimating the kernel density, and the many parameters that
can influence the results in substantial ways when building
more complex representations.

The use of certain distance measures, such as Gower's
distance, allow for missing trait values, with no need for
imputation. In addition, some of the methods for building
NJ trees allow for missing distances between pairs of spe-
cies (Criscuolo and Gascuel 2008). The flexibility offered

by the two methods, i.e. calculating the Gower's distances
and building NJ trees, will help circumvent the issues related
to the many gaps that most trait databases have, particu-
larly for taxa less well studied than birds (Pekar et al. 2021,
Shirey et al. 2022).

The construction of NJ trees is extremely fast, orders of
magnitude faster than hypervolumes, which can be an advan-
tage for large datasets or simulations or null models requiring
many repeated calculations. In addition, it is possible to at least visually estimate to a close approximation many of the metrics derived from tree-like representations (e.g. richness), a task that is much harder for multidimensional representations. This helps avoid errors in data input and/or coding, as many major errors will be obvious through inspection of the tree plot.

## Caveats

The main caveat of using NJ is the lack of apparent connection between trees and the intuitive representation of the Hutchinsonian niche concept (Mammola et al. 2021). It is indeed quite intuitive to imagine the functional space occupied by a group of species as a multidimensional concept depicting its many functional dimensions. Conversely, the connections between species in a tree are not natural in the sense that they do not represent real connections in the community, only the closest path between them in the tree.

A second caveat is a potential lower flexibility than probabilistic hypervolumes to consider the abundances of species in the different metrics. The trait space is largely homogeneous in the way it is occupied, although abundances could theoretically be represented by the density of connections in parts of the tree. In addition, if intraspecific data are available, one can build trees using individuals instead of species, by-passing this issue. Intraspecific trait data are increasingly seen as being crucial to understanding how organisms interact (Tautenhahn et al. 2019, He et al. 2021, Wong and Carmona 2021, Palacio et al. 2024). Given that intraspecific trait data are not always available at the community level, one workaround is to simulate intraspecific variability from compound measures such as the standard deviation of a given trait, which could approximate the kernel-density approach using trees.

## Conclusions

We propose a novel approach to representing functional space and calculating FD that enables the quantification of its different dimensions in ways that combine the strengths of previously proposed FD frameworks. Extensive research on the properties of phylogenetic trees has been undertaken, and, using the NJ framework presented here, these advances can be used in the future to study different properties of ecological systems using functional trees (Ning et al. 2020). The mathematics underpinning NJ are already extensively developed, and thus the use of NJ trees opens up the possibility of testing new hypotheses for FD in the same way as has been done for PD. By combining ease and speed of use, low distortion of functional space, low sensitivity to outliers, and comparability with PD measures, the use of NJ seems a promising approach. More broadly, there are other methods available for building trees that are not considered here, but that could also provide new and advantageous ways to represent functional diversity (Wheeler 2022). As such, we would argue that further exploration and testing of alternatives to

the commonly used functional tree construction approaches (e.g. UPGMA) will likely prove rewarding in the study of functional diversity going forward.

## Author contributions

**Pedro Cardoso:** Conceptualization (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (equal); Writing – original draft (lead); Writing – review and editing (lead). **Thomas Guillerme:** Data curation (equal); Formal analysis (equal); Investigation (supporting); Methodology (supporting); Software (supporting); Writing – original draft (equal); Writing – review and editing (equal). **Stefano Mammola:** Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review and editing (equal). **Thomas J. Matthews:** Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (supporting); Writing – review and editing (supporting). **Francois Rigal:** Investigation (supporting); Writing – original draft (supporting); Writing – review and editing (supporting). **Caio Graco-Roza:** Formal analysis (supporting); Investigation (equal); Writing – original draft (supporting); Writing – review and editing (supporting). **Gunilla Stahls:** Writing – original draft (supporting); Writing – review and editing (supporting). **Jose Carlos Carvalho:** Conceptualization (supporting); Formal analysis (supporting); Investigation (equal); Methodology (lead); Validation (equal); Writing – original draft (equal); Writing – review and editing (equal).

## Transparent peer review

The peer review history for this article is available at https://publons.com/publon/10.1111/ecog.07156.

## Data availability statement

Data are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.c866t1gdw (Cardoso et al. 2024).

## Supporting information

The Supporting information associated with this article is available with the online version.

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