vol. 160, no. 5

the american naturalist

november 2002

Cluster Analysis of Spatial Patterns in Malaysian Tree Species

Joshua B. Plotkin,1,2,* Jérôme Chave,2 and Peter S. Ashton3

1. Institute for Advanced Study, Princeton, New Jersey 08540;
2. Department of Ecology and Evolutionary Biology, Princeton
University, Princeton, New Jersey 08544;
3. Department of Organismal and Evolutionary Biology, Harvard
University, Cambridge, Massachusetts 02138
Submitted February 13, 2001; Accepted May 6, 2002

abstract: Tree species in tropical rain forests exhibit a rich panoplyof spatial patterns that beg ecological explanation. The analysis oftropical census data typically relies on spatial statistics, which quantify
the average aggregation tendency of a species. In this article wedevelop a cluster-based approach that complements traditional spatial statistics in the exploration and analysis of ecological hypotheses
for spatial pattern. We apply this technique to six study species within
a fully mapped 50-ha forest census in peninsular Malaysia. For eachspecies we identify the scale(s) of spatial aggregation and the corresponding tree clusters. We study the correlation between cluster
locations and abiotic variables such as topography. We find that the
distribution of cluster sizes exhibits equilibrium and nonequilibrium
behavior depending on species life history. The distribution of tree
diameters within clusters also varies according to species life history.
At different spatial scales, we find evidence for both niche-based anddispersal-limited processes producing spatial pattern. Our methodology for identifying scales of aggregation and clusters is general; we
discuss the method’s applicability to spatial problems outside of tropical plant ecology.
Subject heading: tropical forests, spatial statistics, spatial point processes, continuum percolation, dispersal limitation.

It has been firmly established that woody plant species andlianas exhibit spatial aggregation in natural communities.
This trend has been especially well documented in speciesrich communities, such as tropical rain forests, throughlarge-scale census experiments (Hubbell 1979; Hubbell
and Foster 1983; He et al. 1997; Condit et al. 2000; Plotkin
et al. 2000b). Although the primary cause of spatial patterns is hotly debated, the existence of spatial heterogeneity
is fundamental to our theoretical and practical under* Corresponding author; e-mail: plotkin@ias.edu.
Am. Nat. 2002. Vol. 160, pp. 629–644. 䉷 2002 by The University of Chicago.
0003-0147/2002/16005-0008$15.00. All rights reserved.

standing of complex ecosystems (Ashton 1976, 1998; Hubbell and Foster 1983; Hubbell 1997; Levin et al. 1997).
The niche-based hypothesis attributes most species-level
heterogeneity to biotic and abiotic environmental variability (Ashton 1969; Grubb 1977). However, even without
habitat differentiation, neutral models of diversity predictthat dispersal limitation can itself account for the emergence of spatial clustering (Wong and Whitmore 1970;
Hubbell 1997, 2001).
Why should tree species be spatially clumped? Why dothey not eventually occupy all suitable space evenly? Many
ecologists (the niche theorists) argue that some sites are,at least at certain periods, unsuitable for certain species.
This occurs because the species is not competitive in thelocal abiotic habitat, or pathogens or predators have eliminated it locally (Janzen 1970; Connell 1971), or it is due
to some random vicissitude. Other ecologists (the neutral
theorists) contend that dispersal limitation and differential
mortality alone cause aggregated occupancy patterns.
Not until recently, however, with the advent of large
botanical inventories and long-term surveys, have thesefundamental issues been addressed with detailed empirical
data. Methodologies for analysis of spatial data have become important for evaluating ecological theories. Suchanalyses usually employ the large body of spatial statistics
that have been developed over the past century. Recent
contributions (He et al. 1997; Batista and McGuire 1998;
Condit et al. 2000; Plotkin et al. 2000b) have quantified
the average clumping characteristics of tree species by utilizing a family of measures derived from the Ripley K statistic (Ripley 1976). The K statistic computes the numberof conspecifics within a distance d from an individual,
averaged over all individuals in the data set. To quantify
clumping, the observed K values are compared with thenull hypothesis of spatial randomness. Other common
methodologies for analyzing spatial data include Fisher’squadrat-based variance-to-mean ratio (Fisher et al. 1922;David and Moore 1954; Lloyd 1967), nearest-neighbor statistics (Pielou 1959; Pollard 1971; Diggle 1983; Cressie
1991; He et al. 1997), and fitted spatial point processes(Diggle 1983; Cressie 1991; Batista and Maguire 1998; Plotkin et al. 2000b).
In this article we address a generic limitation of such

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statistical studies. Despite their broad efficacy, spatial statistics essentially quantify the average aggregation patternof a species. This limitation often has induced ecologists
to assign a single scale of aggregation to each species (Condit et al. 2000; Plotkin et al. 2000b, He and Gaston 2000)
even though a species can be aggregated at several scales
simultaneously. Many ecological questions, such as theinfluence of habitat differentiation, can be better addressedby identifying the specific position, size, and shape of thespatial clusters in a data set. This task is tantamount topartitioning survey data into distinct clusters.
The problem of partitioning data has been investigatedextensively in many disciplines (Duda et al. 1998). In thisarticle we develop an approach to identifying clusters inspatial data, at several spatial scales, without making any
a priori assumptions on the structure of clusters. We apply
this technique to a 50-ha permanent sampling plot of tropical rain forest. Using select species from the 50-ha survey,
we investigate the potential effects of habitat differentiationand dispersal limitation, that is, niche-based and dispersalbased hypotheses, on the spatial arrangement of individuals at various spatial scales. For each species we identify
the scale(s) of spatial aggregation and the corresponding
tree clusters. We study the correlation between cluster locations and abiotic variables such as topography, the distribution of cluster sizes, and the distribution of tree diameters within clusters. We conclude by discussing therelative merits of our cluster-based approach and spatial
statistics in general.
Material and Methods
Study Site
The Pasoh Forest Reserve, Negiri Sembilan, Malaysia, is a2,450-ha protected forest of the Keruing-Meranti type. Its
flora has been studied since 1970 (Wong and Whitmore
1970; Ashton 1976; P. S. Ashton, unpublished manuscript,
1971). The canopy of the Pasoh forest is dominated by
dipterocarp trees, mostly in the genera Shorea, Dipterocarpus, and Neobalanocarpus. Mean rainfall at the studysite is relatively low, approximately 2,000 mm/yr, but without marked seasonality.
A 50-ha permanent sampling plot was established in
1987 by the Forest Research Institute of Malaysia to monitor long-term changes in primary forest (Manokaran and
Kochummen 1987; Kochummen et al. 1990; Manokaran
and LaFrankie 1990). The Pasoh plot, located at 102⬚18⬘W
and 2⬚55⬘N, is a rectangular census 1 km long and 0.5 km
wide. The plot’s topography is fairly even, with a gentle
slope rising 24 m toward the northwest. There are two
primary streams within the plot. All 335,256 free-standing
woody stems exceeding 1 cm diameter at breast height(dbh) were identified to species and spatially mapped to!1 m accuracy (see Manokaran and LaFrankie 1990),
yielding 816 species, 294 genera, and 74 families. The initial census of 1987 has been repeated in 1990 and 1996.
The 1996 data are used throughout this article. The Pasoh
plot is part of a long-term research program worldwide,
coordinated by the Smithsonian’s Center for Tropical Forest Science.
We have selected this plot because it has motivated several statistical studies in the past few years (He et al. 1997;
Condit et al. 2000; Plotkin et al. 2000a, 2000b; He andGaston 2000). Hence, our cluster analysis may be easily
compared with prior results of a more statistical nature.
Study Species
Previous studies of spatial pattern in tropical forests (Heet al. 1997; Condit et al. 2000; Plotkin et al. 2000b) haveaddressed the question of how many species are aggregated. These studies have clearly concluded that most species are clumped, and a few are randomly distributed. Thisconclusion was based on spatial statistics, which are often
deficient for rare species.
Here, by contrast to previous studies, we focus on a
handful of carefully chosen species. We have selected four
abundant and two rare species in the Pasoh permanentplot. Xerospermum noronhianum (Blume) Blume Sapindaceae is a subcanopy species and the most common in
the plot (8,820 stems, average dbh D p 3.19 cm, maximal
dbh Dmax p 37.3 cm). It is recorded as a peat swamp forest
specialist. It is abundant in mesic moist sites throughout
western Malaysia but also is widespread in other lowland
habitats in mixed dipterocarp forest. The seeds of X. noronhianum are dispersed by the monkey, the crab-eating
or long-tailed macaque, and by the gibbon (Yap 1976).
Knema laurina (Blume) Warb. Myristicaceae (4,088 stems,
D p 2.53 cm, Dmax p 35.4 cm) is a subcanopy species
often found on sand or clay.
Neobalanocarpus heimii (3,334 stems, D p 5.0 cm,
Dmax p 196.8 cm) is an emergent dipterocarp and a commercial timber known as chengal. It is endemic to peninsular Malaysia and is widespread there on sandy alluvium and on ridges to about 1,000 m. It is sometimes
gregarious. It is shade tolerant as a juvenile and has very
hard durable wood (specific density 0.7–0.8 g/cm3). Withlarge wingless fruit, the seed has no known means of dispersal. Neobalanocarpus heimii is the ultimate climax species, able to regenerate even in very small light gaps.
Shorea macroptera (1,508 stems, D p 6.1 cm, Dmax p
102.5 cm) is a main-canopy dipterocarp and a leading
species in the so-called Red Meranti–Keruing type of
mixed dipterocarp forest on suitable soils throughout western Malaysia. The fruit bears slightly twisted, aliform, ex-Cluster Analysis of Malay Trees 631
panded sepals, so that it is dispersed by gyration and sometimes wind. It is also a light demander and regenerates
from established seedlings after the formation of large
gaps. Shorea macroptera follows life-historical strategies
that are extremely different from those of N. heimii.
In addition, we have studied two less abundant species
with peculiar spatial patterns. Mallotus penangensis Muell.
Arg. Euphorbiaceae (1,404 stems, D p 4.56 cm, Dmax p
24 cm) is a subcanopy tree found on various habitats bothin peninsular Malaysia and in Borneo (known as Mallotus
sarawacensis). Pternandra coerulescens Jack Melastomataceae (438 stems, D p 5.31 cm, Dmax p 25 cm) is a subcanopy species found mainly along watercourses or wherever ground litter is minimal.

Clustering Spatial Data
Algorithm Specification
We present a technique of spatial analysis that identifies
the location of intraspecific tree clusters in a fully censused
forest plot. This technique will serve as the basis for analyzing the spatial patterns of our six study species. Treelocations are henceforth considered as punctual. Our technical problem is, therefore, to define and detect the spatial
clusters for a set of points in the plane. In addition, wewish to detect “critical distances” or “critical scales” related
to the observed data. When data contain more than one
critical distance, classical statistics would yield an averageof these distances and, thus, a fairly imprecise characterization. The most straightforward method for partitioning
data into clusters is by visual inspection. However, thismethod is subjective and time consuming. We desire arigorous technique of cluster detection that requires no apriori assumptions about the shape of the clusters. We also
want to detect clusters even if they are not radially symmetric or approximately symmetric. We now present anefficient technique for addressing both questions.
We say that two trees are connected if their stems are
separated by a distance d or less. Two trees are said tobelong to the same cluster if they are related by a path ofconnected trees. For example, if tree A is connected totree A, and tree B is connected to tree C, then all threetrees will belong to same cluster, even if A and C are notdirectly connected. This definition of a cluster, illustrated
graphically in figure 1, is extremely simple and natural. In
addition, this definition allows us to interpret cluster analysis in the language of statistical mechanics (see “Percolation and Critical Distances”).
Given a value of the clustering distance d, our algorithm
partitions the data into distinct spatial clusters. The distance d is a free parameter of the algorithm. Once d is

specified, there is a unique corresponding partition of thedata into disjoint clusters. In the trivial case when d p
0, each tree forms its own distinct cluster, but when d isvery large (larger than the maximal pairwise distance between two trees), there is only one cluster; all trees are
pairwise connected. Interesting patterns emerge for intermediate values of the clustering parameter d. The appendix
describes an efficient implementation of our algorithm aswell as some related subtleties of its execution.

Percolation and Critical Distances
Spatial statistics typically are compared against the nullmodel of randomly distributed points. In order to gain
some intuition about our clustering algorithm, we temporarily restrict our focus to the case of random data
points. An understanding of this scenario will improve ourinterpretation of the cluster results in the forthcoming
applied setting. When data are random, the behavior of
our algorithm is, in fact, equivalent to the “continuum
percolation problem,” a well-researched topic in statisticalphysics (Broadbent and Hammersley 1957; Stauffer andAharony 1994). The partitions identified by our algorithm
undergo a sharp transition as the parameter d increases.
Before the transition, most data points are isolated; after
the transition, most points belong to the same large cluster.
This phenomenon is known as the “percolation transition.” Percolation is usually studied when data points are

Figure 1: Schematic spatial arrangement of six data points in the plane.
Points within a radius d are assigned to the same cluster; hence, points
a, b, c, and d are in the same cluster. Points e and f are in separate clusters.

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constrained to remain on a discrete lattice (see, e.g., Keittet al. 1997; and Keymer et al. 2000 for applications inecology). In our case, however, tree locations are unconstrained, and the corresponding process is called “continuum percolation” (Hall 1985; Meester and Roy 1996).
As we vary the cluster parameter d, we obtain corresponding partitions of the data into disjoint clusters. For
any given d value, we denote the number of resulting
clusters by m and the size of the clusters byc 1, c 2 , … , c m. Let n p 冘 ci denote the total number ofindividuals in the sample. We can conveniently summarize
a cluster arrangement by recording the mean cluster size,that is, the first moment of the cluster size distribution:

冘
m

AcS p

Figure 2: a, Idealized relationship between the distance parameter d andthe mean cluster size. A sharp (critical) transition is predicted at a single
distance dc if the data points are randomly distributed in space and if
the sample size is large. As d passes through the percolation thresholddc, the data points become connected in space and the mean cluster size
undergoes a phase transition from 0 to 1. b, Influence of finite sample
size on the “continuum percolation” transition. The transition occurs at
the same critical distance as in the large sample size limit (a), but thetransition is not as sharp. c, Percolation behavior for simulated data. In
each of the four data sets, points have been placed in space at randomwith density 0.1. The largest data set contains n p 12,500 points, and

冘
m

ci
1
# ci p
ci2.
n ip1
ip1 n

The mean cluster size equals the expected size of the cluster
containing a randomly chosen individual. The normalized
mean cluster size, AcS/n, computes the probability that two
randomly chosen data points lie in the same cluster.
Percolation theory predicts that below a certain criticaldistance, dc, individuals will be distributed in many small
clusters, while above this distance most individuals will
belong to a single large cluster. The critical distance dc atwhich the clusters undergo this transition is called the“percolation threshold.” The transition will be increasingly
sharp as number of individuals in the data sample increases. The percolation transition may also be stated interms of the mean cluster size: the (normalized) meancluster size of a very large random data set will equal 0
below the critical distance dc and 1 above the critical distance. Figure 2 demonstrates the percolation transition bygraphing the distance parameter d against the mean cluster
size AcS for a simulated set of random data points.
In real data, as opposed to random data, we do not
expect to find a single sharp transition at one critical distance. Instead, we may often discover several critical distances with varying degrees of sharpness. Such behavior
is indicative of spatial clustering at several scales. As wevary the distance parameter d, producing different partitions of the data into clusters, we will detect critical distances by graphing d against the corresponding mean cluster size AcS. Any range of distances d that corresponds toa plateau in the cluster size curve indicates a nonrandom
scale of aggregation that is insensitive to perturbations in
d. In general, the number of plateaus in the observed

the smallest data set, n p 100 points. As the sample size gets smaller,
the transition becomes more gradual, as depicted schematically in a. See
the appendix for a discussion of minimizing sample size artifacts.

Cluster Analysis of Malay Trees 633
cluster size curve will reflect the number of nonrandom
scales of aggregation.
In order to compare cluster behavior across species withdifferent abundances, we must normalize the distance parameter d. We will divide d by the quantity
1
冑A/n,
2
which is the average nearest-neighbor distance for n randomly placed data points. The normalized parameter isdenoteddˆ p 2d冑n/A.
Here A denotes the total sample area, and n denotes theabundance of the species. This normalization produces anondimensional version of cluster parameter d̂, suitablefor interspecific comparison. In the limit of a large sample
size and a random data set, the sharp transition will occur
around d̂ ≈ 2.4, regardless of the density of points.

Techniques for the Study of Habitat Associations
The association of species with habitat type was one of
the first questions studied by tropical botanists (see, e.g.,
Richards 1936). This question was a primary motivation
for many large-scale studies in Southeast Asia (Ashton1964) and in the Neotropics (Gentry 1988). Such studies
have stimulated the development of statistical methods for
detecting “phytosociological units” or habitat associations.
In essence, statistical methods of habitat correlation
query, with varying degrees of sophistication, whether or
not a species is overrepresented in a particular habitat; x2(Basnet 1992) and modified x2 (e.g., Harms 1996; Plotkin
et al. 2000b) tests are the most common statistical analyses.
Such techniques are undeniably useful in quantifying overall patterns of habitat associations. Yet their ability to characterize the specific effects of habitat on any particularspecies is modest: such tests simply inform us whether or
not a species is associated (positively or negatively) witha given habitat. Moreover, such tests are only possible ifcomplete and consistent abiotic measurements have been
performed throughout the survey area. We will demonstrate how cluster analyses can complement existing statistical techniques. Cluster analysis can elucidate habitat
associations for rare and common species alike, even insituations where more classical statistical analyses fail todetect or to characterize associations. The dual information of the location and the shape of a cluster is extremely
useful in assessing potential habitat specializations.

Results
Identification of Spatial Clumps
As a preliminary example of our cluster algorithm, we firstconsider Mallotus penangensis, a species with a nonisotropic spatial distribution. Figure 3 shows the primary
output of our algorithm. Cluster analyses are shown for
two distances, d p 12.4 m and d p 20.8 m (fig. 3a). Figure 3b shows the relationship between the normalized cluster parameter d̂ and the corresponding mean cluster size.
Notice that the cluster behavior summarized in figure 3b
is very different from the random case discussed above(fig. 2c). Instead of a single percolation threshold at d̂ ≈2.4, M. penangensis displays two transitions occurring at
approximately 1.5 and 8.0. The first transition occurs at
a smaller distance than for random data and the second,at a much larger distance. The distance required to connect
all points into a single cluster is much larger for the Mallotus data (dˆ 1 8) than for the random data (dˆ p 3).
The M. penangensis clusters undergo their first transitionat d̂ ≈ 1.5, and the mean cluster size remains fairly constant on a plateau for a range of distances thereafter (fig.
3b). This indicates that the M. penangensis clusters are welldefined over a large range of scales (1.5 ! dˆ ! 6.0). The
two d values chosen in figure 3a correspond to d̂ p 1.3
and d̂ p 2.2; these two values demonstrate the cluster
structure before and after the first transition. Figure 3b
indicates that the right panel of figure 3a is a better clustering of the M. penangensis species than the left panel, inthe sense that the clusters are robust to perturbation.
The precise relationship between cluster structure anddistance parameter is best summarized by the graph infigure 3b. Nevertheless, figure 4 provides an alternative,
three-dimensional summary of this relationship. The Zaxis of figure 4 corresponds to the distance parameter d.
For convenience, clusters are represented schematically bycircles (in fact, clusters are not symmetric). The top layer,z p 0, shows the locations of every tree in green. On thebottom layer (d p 134 m) all the trees are within the samelarge cluster, represented by a single circle. As d increases
down the Z-axis, the number of clusters decreases and thecluster sizes increase. The critical transitions in this spectrum are visible in figure 4, but they are perhaps discernedmore accurately by graphing the mean cluster size (fig.
3b).
Our analysis of M. penangensis highlights two important
regimes of spatial patterns: stable regions in which theclusters are well defined and sensitive regions in which thecluster structure undergoes dramatic transitions. Depending on the ecological question at hand, either one of theseregimes may be of interest. In the case of M. penangensis,
the stable region most likely indicates the clustering that
corresponds to dispersal-driven clumps of related trees. As

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Figure 3: a, Cluster analysis of the understory species Mallotus penangensis, Euphorbiaceae. There are 1,404 stems of this species in the Pasoh 50ha plot. When we set the clustering distance d equal to 12.4 m, the algorithm partitions the data into 75 distinct clusters, each plotted in a different
color (left panel). Some colors may appear similar. Setting d p 20.8 m results in 34 clusters (right panel). b, Mean cluster size as a function of thenormalized distance parameter for M. penangensis. The algorithm was run with a temperature x p 0.08 to minimize finite-size artifacts (see appendix
for details).

we shall see, however, the critical regimes can also be usefulfor detecting the influences of habitat.

Habitat Associations
Pternandra coerulescens. The species P. coerulescens provides an instructive comparison with M. penangensis. Bothspecies demonstrate essentially two critical distances (figs.
3b, 5b). For P. coerulescens, the transitions occur at d̂ ≈1.2 and dˆ ≈ 2.5. The clusters of P. coerulescens shown infigure 5a occur in the stable region immediately after the

first critical distance. Unlike M. penangensis, the clustersof P. coerulescens are long, narrow, and linear in shape,with a few exceptions. Notice that both cluster geometries
are simultaneously detected by our algorithm. Pternandra
coerulescens also features a large number of singleton trees
located far away from conspecifics. This feature, seengraphically in figure 5a, is reflected by the long tail of thecluster size curve in figure 5b. The tail is caused by thegradual assimilation of singleton trees into the main cluster
at large distances d.
Pternandra coerulescens exhibits clusters with a very
characteristic shape. The hypothesis that P. coerulescens is

Cluster Analysis of Malay Trees 635

Figure 4: Dependence of clusters on the distance parameter d for Mallotus penangensis. The figure shows a three-dimensional graph of M. penangensis
clusters. The X- and Y-axes of the figure correspond to the length and width of the 50-ha plot, as in figure 3a. The Z-axis corresponds to the valueof d. The color also corresponds to the value of d. On the upper plane, d p 0 , and each tree forms its own cluster, all plotted in green. As dincreases down the Z-axis, larger clusters form. Each cluster is represented in the figure by a circle centered at the center of mass of the cluster.
The radius of the circle equals the mean distance of the trees in the cluster to its center of mass.

associated with local streams, which stands out as the mostnatural in light of figure 5b, is confirmed by the natural
history of the species (see “Study Species”). This suggeststhat the linear-shaped clusters are the direct result of P.
coerulescens’s finicky habitat requirements. Indeed, P. coerulescens mainly grows wherever ground litter is minimalor absent. The latter observation was made during a tropical plant biomass experiment in 1971–1973 when all freestanding trees and litter were removed from a survey plotnear Pasoh’s 50-ha plot (Kira 1971). Pternandra coerulescens, normally a fairly rare species, colonized most of thelogged region, presumably due to the complete absence ofground litter there (T. Kira, personal communication). In
addition, P. coerulescens also clusters at the primary pig
wallows in the plot (fig. 5a), another location with frequent
litter disturbance. Despite the numerous, widespread,small seeds produced by P. coerulescens, the species is severely limited by abiotic environmental factors. Since mapsof Pasoh’s local stream networks (and many other abiotic
environmental variables) are not available, the present
analysis would have been much more difficult, or impossible, via classical tests of habitat specificity.
Xerospermum noronhianum and Knema laurina. We turn
now to the environmental influences on two of the mostcommon Pasoh species: Xerospermum noronhianum andKnema laurina. In particular, we examine the effect of
topography of their spatial distributions. A Ripley K analysis indicates that both of these species follow nonrandom
spatial distributions (Plotkin et al. 2000b). The cause or
nature of the nonrandomness is difficult to determine with
the naked eye. Statistical tests indicate that neither of these636

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Figure 5: a, Cluster analysis of Pternandra coerulescens that results fromthe distance parameter d p 25.4 m (dˆ p 1.5). There are 438 stems within
the plot, which form 79 clusters. Most of the clusters of P. coerulescens
are linear in shape. The arrow indicates the locations of pig wallows
within the 50-ha plot. b, Mean cluster size curve for P. coerulescens. The
algorithm was run with a positive temperature (x p 0.08 ) to minimize
finite size effects. The resulting curve shows two principal transitions.

species is very strongly correlated with Pasoh’s topography(Plotkin et al. 2000b). In particular, for each species, thetotal density of stems on the hill is roughly equal to thedensity of stems off the hill. Although topography does
not limit the ranges of these species, it might, nevertheless,influence the spatial geometry of individual treeplacement.
Figure 6b shows that topography does have an effect on
the spatial distribution of these two species, despite theabsence of a strong statistical correlation with topography.
Knema laurina exhibits one large cluster whose boundaries
clearly coincide with the detailed topography of the single

Figure 6: a, Topographic contour map of the Pasoh 50-ha plot. The
primary water drainage route running off the hill is indicated in red. b,
Cluster analysis of Knema laurina for d p 12.1 m (330 disjoint clusters).
Notice that K. laurina forms a single, stable cluster on the hill. c, Cluster
analysis of Xerospermum noronhianum for d p 9.0 m (494 disjoint clusters). Notice that X. noronhianum tends to form larger, stable clusters
off of the hill. The water drainage route apparently disrupts the X. noronhianum clusters. See figure 7 for the relationship between distanceparameter d and mean cluster size.

Cluster Analysis of Malay Trees 637
Cluster Demographics
The ability to partition spatial data into clusters allows us
to reexamine many classical questions about plant community dynamics, such as habitat associations. We can also
now analyze community structure on a different scale thanthe census scale addressed by classical statistics. In particular, having identified specific tree clusters, it is natural toconsider the range of observed cluster sizes as well as thewithin-cluster demographics.

Figure 7: Mean cluster size curve for Xerospermum noronhianum. The
algorithm was run with a positive temperature, x p 0.05 (see appendix
for details). The critical value d̂ p 2.4 yields the cluster arrangement
displayed in figure 6c. The mean cluster size curve for Knema laurina(not shown) is very similar.

hill within the Pasoh plot. Knema laurina’s cluster geometry is radically different off the hill as compared toon the hill. Off the hill, K. laurina forms tighter clumps
that are more separated from each other than those on
the hill. Since the total densities of trees on and off thehill are roughly equal, the within-clump density of stems
is higher for clusters off the hill, an intriguing behavioral
pattern whose underlying mechanism we cannot yet fully
elucidate.
Although the cluster behavior of X. noronhianum is lessdramatic than that of K. laurina, it also evidences a clear
response to the topographic gradient. In the case of X.
noronhianum, the clusters off the hill are large, stable, andevenly spaced. On the hill, X. noronhianum exhibits tighter
aggregation patterns. In fact, the cluster geometry of X.
noronhianum is also correlated with the main drainage
streams running off the hill (as indicated in fig. 6c). Figure
7 shows the cluster size curve for X. noronhianum. The
effect of topography on cluster geometry has been revealed
by inspecting a critical value of d̂ p 2.4 (d p 9.0 m).
As figures 6 and 7 reveal, the spatial cluster geometriesof X. noronhianum and K. laurina are clearly influenced
by topography. It is unlikely that these species suffer froma dramatic difference in dispersal limitation depending on
their location on or off the hill. Thus, habitat is probably
the primary influence on the particular spatial arrangement of these two common species, despite their constitutive presence throughout the plot.

Cluster Sizes. Null models of disturbance-driven communities would predict uniform cluster sizes within eachspecies, reflecting the same history of disturbance. As isfairly clear from figures 3–7, there is a very wide range ofcluster sizes within each species, both in terms of the minimum convex area containing a cluster and the numberof trees belonging to each cluster. Figure 8 summarizes
the distribution of cluster sizes (in terms of number ofindividuals) for M. penangensis, P. coerulescens, and K. laurina. Each of these species, as well as the vast majority ofPasoh species, demonstrates a wide range of cluster sizes,
even if we discard extraneous clusters containing ≤5individuals.
Distribution of Within-Cluster Tree Diameters. We now focus on the within-cluster demographics of our study species. Figure 9 shows the range of tree diameters withineach of the four largest clusters of M. penangensis. For
comparison, we also plot the dbh distribution of all M.
penangensis stems in the entire 50-ha plot. Aside from thefour or five largest trees, the clusters exhibit the same fullFigure 8: Distribution of cluster sizes for Mallotus penangensis (dashes),
Pternandra coerulescens (solid line), and Knema laurina (dots). The figure
shows the rank cluster size distribution on semilogarithmic axes. All three
species demonstrate a full range of cluster sizes. (The largest cluster ofK. laurina, containing 2,223 trees, is not visible within these axes.)

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Figure 9: Distribution of tree diameters within the four largest Mallotus
penangensis clusters, compared with the distribution of M. penangensis
diameters in the entire 50-ha census (large dots). The figure shows a rank
dbh plot on a semilogarithmic axis. The individual clusters of M. penangensis exhibit nearly the same full range of dbh classes as the 50-haplot as a whole.

range of diameters as the 50-ha census taken altogether.
Moreover, the dbh distribution is approximately exponential at both the cluster scale and the full 50-ha scale.
These results indicate that the identified clusters of M.
penangensis represent full-fledged microcommunities oftrees, containing juvenile and mature stems alike. We do
not find any examples of purely juvenile or purely adulttrees, which would be obvious signatures of a system awayfrom equilibrium. Instead, the clusters of M. penangensis
are in equilibrium, in that we do not expect their diameter
distribution to change significantly in time.
A full range of intracluster diameters is common among
those Pasoh species that demonstrate obvious aggregation
patterns. The trend demonstrated by M. penangensis also
holds for our other study species. Species Neobalanocarpus
heimii is a rare counterexample: its large clusters demonstrate an elevated proportion of juveniles, comparedwith the dbh distribution in the plot as a whole. The
unusual nonequilibrium distribution of N. heimii reflects
its characteristic ability to regenerate quickly in small gaps.
Cluster Structure and Tree Diameters. In addition to therange of within-cluster tree diameters, we are also interested, naturally, in the relative locations of large and small
trees within a cluster. In particular, are clusters characterized by one or several large “mother trees” located near
the cluster center? Or are large trees more often found on
the periphery of clusters? Such questions are relevant toan understanding of those processes behind cluster
formation.

We have found examples of both trends within the Pasoh plot. Many species exhibit random within-cluster treeplacement, with respect to tree diameter. Many other species exhibit an elevated proportion of large peripheral
trees. And only a few species exhibit large central trees.
This finding, which we document in two particular instances below, may be somewhat surprising compared withthe simplistic view of a dispersal-driven cluster with central
large “mother trees.” Shorea macroptera exhibits an elevated proportion of peripherally located large trees. Figure
10a shows a cluster analysis of S. macroptera in which thesize of each circle corresponds to the tree’s diameter. It is
not immediately apparent to the eye whether large trees
are central or peripheral. With the explicit knowledge ofthe cluster partition, however, we can test for statisticalcorrelations between tree diameter and within-cluster
placement.
We define the center of a cluster to be the average X
and Y coordinate of all those trees in the cluster. Thisdefinition assumes some degree of isotropy. For each cluster, we inspect the relationship between a tree’s dbh andits distance to the center of its cluster. We normalize thisdistance by dividing by the mean distance to the center
among all trees in the cluster. Finally, we calculate thePearson’s correlation coefficient of the relationship between dbh and distance to center for each cluster.
The correlation between tree diameter and distance tocenter is shown for each cluster of S. macroptera in figure
10b. Notice that most of the large clusters demonstrate apositive correlation. Pooling the dbh distance data fromthe 10 largest clusters of S. macroptera together, the positive
correlation between dbh and distance is statistically significant at the 95% confidence level: larger trees tend tolie on the periphery of their cluster. The same finding holds
for many other Pasoh species, including, for example, Dipterocarpus cornutus, Shorea guiso, Schoutenia accrescens,and Eugenia claviflora.
Figure 11 demonstrates a parallel analysis for N. heimii.
In this case, however, the large clusters demonstrate a negative correlation between dbh and distance to cluster center. Pooling the data from the 10 largest clusters together,
there is again a statistically significant result at the 95%
level: larger trees tend to be found closer to the center ofthe cluster. Among the Pasoh species, N. heimii is a somewhat rare example of this central large tree phenomenon.
The tendency for large trees to be located in the center ofclusters, as opposed to the periphery, is indicative of strong
intraspecific competition. Despite an abundant seed rainaround many large, reproductive, locally dispersed trees,
seedlings have difficulty establishing themselves in the
nearby vicinity of large conspecifics and find themselves
marginalized on the periphery. Most of this intraspecificcompetition occurs before stems reach 1 cm diameterCluster Analysis of Malay Trees 639
reproductively for centuries. Because of their large winglessfruit, most juveniles do not survive the stifling dark parental subcanopy light climate. As a result, there are several
large central stems of N. heimii surrounded by a range ofaspiring juveniles on the perimeter, most of which will notsurvive. This interpretation agrees with our earlier observation of an elevated proportion of juveniles found inclusters of N. heimii.
The seeds of S. macroptera, by contrast, are well dispersed, and the growth rates are also higher. Moreover,
adult stems of S. macroptera do not persist as long as N.
heimii. As a result, old clumps of S. macroptera may havenew clumps growing in gaps and around their perimeters;
we therefore find a positive correlation between dbh anddistance to center.
It is also interesting to compare the cluster demograph-

Figure 10: a, Cluster analysis of Shorea macroptera corresponding tod̂ p 1.4, resulting in 171 clusters. The radius of each circle represents
the diameter of the corresponding tree, although not to scale. b, Pearson
correlation coefficient for the within-cluster relationship between dbh
and distance to cluster center for all S. macroptera clusters. The X-axis
denotes the size of the cluster, and the Y-axis denotes the correlation
coefficient for the trees within the cluster. Notice that most of the large
clusters exhibit a positive correlation between dbh and distance to center;large S. macroptera trees are found toward the periphery of their cluster.

(Wills and Condit 1999). Thus, within-cluster demographics may be a strong indicator of intraspecific forces andeven Janzen-Connell effects (Janzen 1970; Connell 1971).
It is interesting to note that N. heimii and S. macroptera
have radically different dispersal mechanisms (see “Study
Species”). The dispersal differences between these two dipterocarps are clearly correlated with the different spatial
population structures of their clusters (fig. 10 vs. fig. 11).
Such an observation would perhaps be predicted from anunderstanding of life-history traits and from visual inspection but can only be verified rigorously by a complete
cluster analysis. In the case of N. heimii, trees that succeed
to maturity then escape competition and continue to live

Figure 11: a, Analysis of Neobalanocarpus heimii parallel to figure 10.
The cluster algorithm was run at d̂ p 1.1, resulting in 1,200 clusters. b,
Pearson correlation coefficient for the within-cluster relationship between
dbh and distance to cluster center for all N. heimii clusters. Notice thatmost of the large clusters exhibit a negative correlation between dbh anddistance to center. In other words, large trees of N. heimii are found near
the center of their cluster.

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ics of N. heimii with those of the ballistically dispersed M.
penangensis. On the face of it, these two species might be
expected to demonstrate similar within-cluster spatial
structure. However, M. penangensis does not, in fact, exhibit any significant correlation (positive or negative) between distance and size, in contrast to N. heimii’s negative
correlation. This observation likely reflects the fact that N.
heimii does not reproduce until its crown has emerged
above the canopy, perhaps after at least 50 yr. Mallotus
penangensis, which also is a slow grower, nevertheless starts
reproducing when quite small, so that a range of sizes are
dispersing seeds within the cluster, resulting in no consistent space-diameter correlations.
In summary, we have seen that there is a large range ofcluster sizes within each species and that clusters generally
exhibit a full range of tree sizes, whose distribution issimilar to the dbh distribution found in the 50-ha plot as
a whole. In addition, many species exhibit large trees on
the periphery of their clusters, few species exhibit clusters
centered around large trees, and many species exhibit noconsistent spatial size class structure. And finally, many ofthese trends reflect the different dispersal mechanisms andgrowth patterns between species.

Discussion
Ecological Implications
Ecologists address questions of community dynamics on
many spatial scales: individuals, censuses, regions, andlandscapes. Once spatial data have been partitioned into
clusters, however, we can inspect community dynamics on
two new scales: the intracluster and intercluster scales. In
our case, clustering has allowed us to examine several issues of importance in tropical plant ecology: the correlation between cluster locations and abiotic environmental
variables, the distribution of cluster sizes, the distribution
and spatial placement of within-cluster tree diameters.
This technique has allowed us to unveil hidden patterns(for Xerospermum noronhianum and Knema laurina)
whose discovery might otherwise have been impossible.
Such discoveries suggest that habitat may have a stronger
effect on the spatial distribution of common species—even
species that are present constitutively on all habitat
types—than standard statistical analyses would suggest. In
addition, cluster analysis of species such as Pternandra
coerulescens demonstrate that spatial distribution cansometimes be driven almost entirely by abiotic habitat
specificity, regardless of dispersal capability.
At the same time, our investigation of cluster demographics has revealed that dispersal mechanisms andgrowth patterns can play a large role in determining intracluster spatial population structure. Species-level variation in regeneration speed is reflected by variation in therange of tree diameters found within clusters. Gap colonizers (e.g., Neobalanocarpus heimii) may be identified,from static data alone, by a signature elevated proportionof purely juvenile clusters. Differences in dispersal mechanism (e.g., Shorea macroptera vs. N. heimii) are associatedwith characteristic differences in the placement of large
trees within a cluster. Similarly, differences in growth patterns (e.g., Mallotus penangensis vs. N. heimii) are also
reflected by within-cluster tree placement; species that reproduce throughout their lifetime may lack any correlation
between diameter and intracluster placement.
In short, we have found that the diversity of tropicaltree life-historical traits and of the diversity of responses
to abiotic influences are reflected by the diversity of their
spatial arrangements. Thus our study species evidence bothniche-based and dispersal-based processes affecting spatial
distributions, at different spatial scales.
We emphasize that, despite the analysis of our study
species, our cluster method does not allow us to inferprocess unambiguously from pattern. We cannot soundly
argue to have teased apart the influences of niche- anddispersal-based processes in general. In the end, we viewthe techniques developed here more as tools for exploratory data analysis and for generating hypotheses. Suchhypotheses (e.g., the disequilibrium dynamics of N. heimii)
must eventually be tested by field and lab experiments.
Nevertheless, we have seen that careful direct observationof species’ maps can, when combined with a clustering
algorithm, provide a powerful method for ecological
exploration.

Cluster Analysis and Spatial Statistics
We have developed a fairly general method for analyzing
spatial distributions that is conceptually different from theclassical paradigm of spatial statistics. Spatial statistics havebeen used to assess mean clumping behavior across theentire range of species in order to document universal
trends of aggregation in forests. The cluster techniquesdeveloped here complement such gross statistics.
One might argue that a formal clustering technique provides little more information than an automated mapping
device. The human eye is, in fact, a remarkably efficientand complex tool for detecting spatial patterns. The methodology developed here, however, permits us to reframe
specific questions of tree dynamics in terms of spatial scales
and to identify the relevant critical scale(s) for a given
species. Several critical scales indicate multiple factors that
influence a species’ distribution. There is no single “correctCluster Analysis of Malay Trees 641
answer” to partitioning spatial data but, rather, a spectrumof answers among which our method guides us toward
the critical and stable solutions.
Unlike most spatial statistics, our method of identifying
clusters does not make a priori assumptions about underlying spatial structure. The edge-correction factors used
to compute spatial statistics, by contrast, typically assume
that the underlying stochastic point process is isotropic or,at least, stationary (Ripley 1976). We have seen, however,that abiotic influences often cause large departures fromstationarity or isotropy in tropical tree distributions. Departures from these assumptions are also common in otherspatial settings (Diggle 1983).
The primary distinction between cluster analysis andspatial statistics is that the former explicitly separates data
points into disjoint subsets. Formal clustering is betterused as a tool for exploratory data analysis of the specificspatial geometries in a census; spatial statistics, by contrast,
are more limited and require more assumptions but allow
for rigorous statistical tests of departure from spatial randomness (He et al. 1997; Condit et al. 2000; Plotkin et al.
2000b).
Unlike gross statistics, the identification of explicit clusters allows for further inquiry into the spatial geometryof a data set. In our case, clustering has allowed us to
examine several questions of ecological importance: thecorrelation between cluster locations and abiotic environmental variables, the distribution of cluster sizes, and the
distribution and spatial placement of within-cluster tree
diameters. Such analysis of specific ecological and demographic questions in light of the known life-historical strategies would be impossible without partitioning the data
set into clusters.
Our method for identifying scales of aggregation andclusters is general, and it can be applied in contexts otherthan tropical plant ecology. For example, spatiogenetic
analysis of plants in general may benefit from a formal
clustering methodology to complement statistical techniques that are currently employed (Ouborg et al. 1999).
In that setting, such a methodology would allow researchers to contrast the genetic variation within and between
clusters. Our method of identifying scales of aggregation
can also be applied outside of a strictly spatial setting, forexample, for the identification of quasi species within adata set of RNA virus sequences (Plotkin et al. 2002).

Acknowledgments
We are most indebted to N. Manokaran for allowing us
access to the Pasoh 50-ha data set. Thanks are also dueto the Smithsonian’s Center for Tropical Forest Science.
We thank E. Domany, who provided us with his superparamagnetic clustering algorithm for comparison with
our algorithm. We also thank H. Muller-Landau, whopointed out the connection with the single-link algorithm.
J.B.P. acknowledges support from the National Science
Foundation, the Teresa and H. John Heinz III Foundation,and the Burroughs Wellcome Fund. J.C. was supported by
the Andrew Mellon Foundation and the David and Lucile
Packard Foundation (99-8307) through grants to Simon
Levin.

APPENDIX

Algorithmic Details
In this appendix we detail the algorithmic implementationof our data clustering method.

Algorithmic Implementation
Our basic clustering algorithm is extremely simple. We
first fix the clustering parameter d, and we connect twotrees if their pairwise distance is less than d. Next, we
identify the resulting connected components (clusters).
The latter stage is critical from a computational viewpoint.
A naive implementation of this step would involve a double loop over all the data points. For n trees, the algorithm
would require on the order of n2 operations. An efficient
alternative for identifying connected components has beendeveloped by Hoshen and Kopelman (1976), and it requires n ln (n) operations.

Minimizing Sample Size Effects
Our basic algorithm can be modified to reduce the noise
associated with finite sample sizes (as seen, e.g., in fig. 2c).
In the original algorithm, we first fix the clustering parameter d, and we connect two trees if their pairwise distance is less than d. In the modified algorithm, we assume
a small amount of noise in the parameter d, and we averageover many replicas of the basic algorithm. The noise issomewhat analogous to temperature in statistical mechanics. For a fixed mean value of Ad S, we compare pairwise interevent distance to independent draws of d froma normal distribution with mean Ad S and standard deviation x # Ad S (the normal distribution is naturally truncated at d p 0). Our simplest algorithm corresponds tothe deterministic case x p 0. When x is positive, however,
the algorithm is no longer deterministic. For each replica,
the radius around each data point varies slightly. We first642

The American Naturalist

fix Ad S, and we average the resulting mean cluster size overmultiple runs of the stochastic algorithm. This averaging
procedure reduces the finite-size artifacts. Empirical investigations suggest that x ≈ .05 Ⳳ .03 is a good range ofvalues for minimizing noise in ecological data.
As an aside, we mention one important technicality regarding the stochastic version of our algorithm. The criticaldistance dc studied extensively in physics and mathematics
depends on the value of x. In fact, simulations suggestthat as x ranges from 0 to 0.25, the normalized criticaldistance of the Poisson process varies from to 2.44 to 2.18.
Fortunately, a result of Meester and Roy (1996) guarantees
that dc depends continuously on x. (Meester and Roy[1996] prove that if two series of distributions for the radii
converge weakly, then their corresponding critical distances converge as well.) In most theoretical studies of
percolation, x is assumed to be 0.

Other Clustering Algorithms
There exists a large variety of clustering algorithms (kmeans, fuzzy k means [Duda et al. 1998], neural k means[Kohonen 1989], nearest neighbor, furthest neighbor, centroid, Ward, hypervolume, minimal spanning tree, Mojena’s upper tail, Wolfe’s test) designed to partition data
into clumps (Everitt 1993). Many of these techniques require explicit or implicit a priori assumptions about cluster
shapes or the total number of clusters. In an extensive
comparison of clustering techniques, Hardy (1996) concluded that the implicit assumptions of most algorithms
often lead to erroneous cluster classifications. Our methodof clustering is closely related to the so-called single-linkage hierarchical method. Our method is distinguished,however, by the introduction of a temperature, or statistical averaging, into the cluster calculation. Moreover, theconnection between single-linkage clustering and continuum percolation provides an important tool for analyzing
the spectrum of possible cluster arrangements and for
identifying either the critical or the stable solutions.
A new generation of clustering algorithms, also inspiredby percolation theory, has recently appeared in the physics
literature (Rose et al. 1990; Blatt et al. 1997; Angelini etal. 2000). The unifying concept of these methods is to
optimize the temperature variable x for each value of theaverage distance Ad S. Statistical mechanics can be used tosolve this problem in a broad class of dynamical systems(coupled map lattices and superparamagnetic systems),
and the resulting algorithm is, therefore, powerful and, ina sense, optimal. We have verified that one of these algorithms (SPC, provided by E. Domany) gives very similar
results to our more simple algorithm. We believe that therelative simplicity of our algorithm is valuable in the present context.

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