International Journal of Forestry Research

International Journal of Forestry Research / 2017 / Article

Research Article | Open Access

Volume 2017 |Article ID 2818132 |

Lurdes Borges Silva, Mário Alves, Rui Bento Elias, Luís Silva, "Comparison of -Square, Point Centered Quarter, and -Tree Sampling Methods in Pittosporum undulatum Invaded Woodlands", International Journal of Forestry Research, vol. 2017, Article ID 2818132, 13 pages, 2017.

Comparison of -Square, Point Centered Quarter, and -Tree Sampling Methods in Pittosporum undulatum Invaded Woodlands

Academic Editor: Guy R. Larocque
Received01 Aug 2016
Accepted15 Nov 2016
Published11 Jan 2017


Tree density is an important parameter affecting ecosystems functions and management decisions, while tree distribution patterns affect sampling design. Pittosporum undulatum stands in the Azores are being targeted with a biomass valorization program, for which efficient tree density estimators are required. We compared -Square sampling, Point Centered Quarter Method (PCQM), and -tree sampling with benchmark quadrat (QD) sampling in six 900 m2 plots established at P. undulatum stands in São Miguel Island. A total of 15 estimators were tested using a data resampling approach. The estimated density range (344–5056 trees/ha) was found to agree with previous studies using PCQM only. Although with a tendency to underestimate tree density (in comparison with QD), overall, -Square sampling appeared to be the most accurate and precise method, followed by PCQM. Tree distribution pattern was found to be slightly aggregated in 4 of the 6 stands. Considering (1) the low level of bias and high precision, (2) the consistency among three estimators, (3) the possibility of use with aggregated patterns, and (4) the possibility of obtaining a larger number of independent tree parameter estimates, we recommend the use of -Square sampling in P. undulatum stands within the framework of a biomass valorization program.

1. Introduction

In ecological research, the basic objective of sampling is to obtain a descriptive estimate of some attribute of a plant population. This estimate should be a relatively accurate representation to allow detection of real differences among plant populations. Quantitative data are essential to adequately characterize the woody component of forest communities [1]. Some form of sampling is required because total counts of individuals in naturally occurring plant populations are generally impractical without an exhaustive expenditure of energy and resources [2]. A number of sampling techniques are available to quantify forest community traits. These techniques vary in their underlying assumptions (e.g., random distribution of target population), equipment required, and time necessary to obtain an adequate sample for statistical analysis [3].

One of the most commonly sampled parameters is density [4]. Stand density or tree density, expressed as the number of trees per unit area, is an important forest management parameter. Together with other forest structure parameters such as crown closure and crown diameter, it is used by foresters to evaluate regeneration, to assess the effect of forest management measures or as an proxy variable for other stand parameters such as age, basal area, and volume [5]. Thus, density is considered as one of the most important numerical indices to explain quantitative values of tree and shrub communities. By affecting many aspects of the ecosystem, it can be used in a wide variety of situations.

One of the oldest techniques of data collecting is quadrat counting [6] but this analysis may be insufficient to distinguish certain point patterns and the size of quadrats may affect the results [7]. Obtaining adequate information with minimum effort and time is a major concern when sampling vegetation [3]. In sampling for indices of forest structure, distance sampling (plotless sampling) can be very efficient [8]. Continued technological advances in range measurements and field computing add to the attraction of distance sampling because measurements can be completed ever faster. Also, in natural populations characterized by an irregular, possibly clustered, distribution of trees, the precision of a density estimate from distance sampling can be better than the precision obtained with fixed area plot sampling [9]. Distance sampling has attracted the attention of researchers over the past 50 years as a means of estimating density. Its main attraction is that it is fast and easy to use, and one or more distances are always recorded at each sample point. In contrast, plot sampling can sometimes be a very time consuming process, boundary trees may be overlooked, and some plots may have no tallies [10]. Distance sampling has a long history in forest inventories where the distance and attributes of interest are measured on the -trees closest to a sample point [9]. When quadrat sampling is difficult or too costly (e.g., in low density populations or mountain areas) distance sampling is favored [11]. Distance sampling is often faster than fixed area plot sampling, since the number of measurements to take at each location can be fixed in advance. This is because there is no need to sample all the trees present in a plot and therefore measurement efforts are independent of local tree density. Expediency that is faster data collection in stand-wise forest management inventories has been another motivating factor [12].

A variety of methods using distance measures have been developed [1]: -Square sampling [13, 14], Point Centered Quarter Method [15], -tree sampling [16], random pairs [17], variable area transect [18], closest individual [15], ordered distance [19], compound estimators [20, 21], corrected point-distance [22], nearest neighbor [15], quartered neighbor [23], and angle order [24]. All these methods are essentially based on the distance measures from event to event or from point to event [1, 25].

A variety of density estimators have been proposed for different spatial patterns and modifications to improve their robustness, especially when nonrandom spatial patterns are assumed [15, 20, 24, 27]. Some of the distance methods can be used to determine the general spatial pattern or distribution of a population [13, 28] and are useful to determine inter- and intraspecific relationships in plant communities [29]. Advantages arise from ease of use calculation and interpretation by foresters [25]. Their great disadvantage is connected with the loss of detailed information about spatial patterns at different spatial scales [30], describing the spatial structure only at the fine scale of nearest neighbors [25].

Forest stand structure is a key element in understanding forest ecosystems and many methods have been developed for its study [31]. One of the major components of forest stand structure is the spatial arrangement of tree positions [8]. Spatial information for individual trees is increasingly sought by forest managers and modelers as a means to improve the spatial resolution and accuracy of forest models and management scenario [32]. There are three general categories of two-dimensional point patterns, uniform (hyperdispersed), random, and clumped (aggregated). Random type of spatial distribution means that trees are distributed independently of each other and the probability of finding trees in the whole population is the same. In aggregated populations, individuals occur in clumps of different densities and sizes, and in a regular distribution objects are evenly spaced in a population over a given area [25]. We can assume that the spatial pattern reflects the major trends in stand dynamics. For example, regular spatial structures indicate high competition, whereas aggregate patterns indicate massive regeneration without subsequent strong self-thinning [29].

According to Elias et al. [33], in Azorean native mountain forests, the spatial pattern of tree species is largely explained by disturbance regimes and the regeneration strategies of each species. However, factors such as habitat related patchiness, competition, and dispersion limitation may also explain many of the observed patterns. Meanwhile, spatial patterns in Azorean exotic woodland have not been studied. Invasive alien plants are one of the most relevant threats to the long term integrity of biodiversity, affecting the Azores archipelago, with a relatively high rate of nonindigenous species, among which are several top invaders [3436]. Pittosporum undulatum Vent. (Pittosporaceae) is one of the top invasive trees in the Azores, affecting protected areas and indigenous species, and was targeted by the Azorean Regional Program for Control and Eradication of Invasive Plants in Sensitive Areas (PRECEFIAS) [3640]. Accurate data about abundance and distribution of P. undulatum is needed for its control and management, including the evaluation of the energetic potential of its biomass [38, 41]. This information will be crucial for the implementation of a large scale survey of P. undulatum stands in the Azores.

Therefore, the aims of this study are (i) to evaluate the most accurate and precise method to estimate tree density in Azorean exotic woodland; (ii) to test plotless density estimators that have been recently validated for other species and regions; (iii) to compare results with those from quadrat counts; (iv) to determine tree distribution pattern in exotic woodland.

2. Material and Methods

2.1. Study Area

We investigated P. undulatum dominated forests located in São Miguel Island (Azores) situated in the North Atlantic Ocean, 1500 km west from Portugal (Figure 1). The archipelago of the Azores, scattered across 615 km on a WNW–ESE alignment, covering a total of 2323 km2, comprises nine volcanic islands. The climate is temperate oceanic with a mean annual temperature of 17°C at sea level. Relative humidity is high while rainfall ranges from 1500 to 3000 mm, depending on the altitude and increasing from east to west [42]. The topography of the island is characterized by volcanic craters, large catchments, ravines, and seasonal water streams with a maximum altitude of 1105 m above sea level and a surface area of 745 km2.

2.2. Stand Characterization

The studied stands (Figure 1) are located within geomorphological formations [43], namely, Nordeste Volcanic System and Povoação Volcano (Plots 1 and 2), Sete Cidades Volcano (Plots 3 and 4), and Picos Fissural Volcanic System (Plots 5 and 6). Owing to their volcanic origin, soils are generally young Andosols, formed under a humid climate on relatively recent lava flows and pyroclastic deposits. Moreover, Ribeira do Guilherme is a protection area for the management of habitats or species and Pico das Camarinhas is a geological monument [44]. The characterization of each plot is presented in Table 1.

Plot numberLocationMaximum elevation (m a.s.l.)Area (ha)Other species

1Ribeira do Guilherme700120.0Laurus azorica, Morella faya

2Mata dos Bispos5002.5Laurus azorica, Erica azorica,
Clethra arborea

3Pico das Camarinhas25010.0Morella faya

5Pico da Furna3112.2Acacia melanoxylon

6Picos de Lima,
Fenais da Luz
3122.3Acacia melanoxylon

West slope, outside volcanic crater, and Caldera, inside volcanic crater.
2.3. Plot Sampling

The field work was carried out in May and June 2014. At each stand we marked a plot with  m (900 m2). We divided each plot into 36  m quadrats. At each quadrat, the total number of trees was thoroughly counted. Saplings less than 2 cm of diameter at breast height were excluded because most likely they would not survive. All woody species that were within the plots were included in the measurements.

Regarding plant density obtained with plot counts we estimated the confidence interval using the usual formula for the mean:where is the mean density and is the respective standard error.

2.4. Plotless Sampling

Thirty random points were selected at each plot. Distances from point-tree and tree-neighbor were measured at 1.30 m high, at the center of each tree. Three methods were used.

In -Square [1, 13, 14, 20, 26], Figure 2(a), a random point is chosen and the distance to nearest tree is measured. A second distance is measured from the nearest tree to its nearest neighbor constrained to be in the hemisphere to the left of the dashed line. Four density estimators were used (Table 2). For -Square sampling the 95% confidence interval for the reciprocal of tree density is calculated aswhere SE is the standard error given by Diggle [31], is the density estimator for -Square (Table 2), and are the respective distance means, is the variance of , is the variance of , is the covariance of and , and is the value taken from a Student distribution with degrees of freedom.

Sampling methodsEstimatorsCharacteristic

-Square[13]λ = estimated density
= sample size
= distance from closest individual to nearest neighbor
= distance from random point to closest individual

Point Centered Quarter[15] = estimated density
= sample size
= number of individuals to be located in each sector of the area around the random sampling point
= distance from random point to closest individual in quarter

-treeFor 3-tree = sample size
= estimated density
= plot area in each plot is calculated separately using the following equation:

= plot radius and in normal method prolongs to mid diameter of th tree but in adjusted methods prolongs between and trees.
3-tree adjusted
For 4-tree
4-tree adjusted
For 5-tree
5-tree adjusted
For 6-tree
6-tree adjusted [16]

In Point Centered Quarter Method (PCQM; [15, 19, 24]; Figure 2(b)), the area around the random point is divided into four 90° quadrants and the distance to the nearest tree is measured in each quadrant. Three density estimators were used (Table 2). For PCQM confidence limits are given by Seber [45]:where is the density estimator for PCQM (Table 2), is the distance from random point to closest individual in quarter j (; ), and is the value taken from the standard normal distribution, for an error .

In -tree sampling (-tree sampling; [16, 27]; Figure 2(c)), a number of trees, N, closest to a random point are selected and the respective distances measured. Plot shape is circular and is based on a radius from the sampling point to the centre of the th tree closest to the point. In our study we considered a value of (see Table 2 for estimators).

2.5. Spatial Pattern Analyses

Since the 1950s, several spatial methods of analysis have been developed and modified to improve our ability to detect and characterize spatial patterns [46]. We used five indexes of spatial pattern, applied to our data from -Square [47, 48], PCQM [28, 49], and quadrat sampling [50].

For -Square we used the index recommended by Ludwig and Reynolds [48]:The value of C is approximately 0.5 for random patterns, significantly less than 0.5 for uniform patterns, and significantly greater than 0.5 for clumped patterns. To test the significance of any departure of C from randomness, we compute a value for (see [48], pp. 57-58 for details). We also used the statistic recommended by Hines and O’Hara Hines [47]:This test statistic is compared with reference values (see [6], pp. 210 for details). Randomness corresponds to 1.27; smaller values indicate a uniform pattern while larger values indicate clumping.

We used our data from PCQM by selecting the tree closest to the sample point [6, 25, 48] and using the index of dispersion described by Johnson and Zimmer [49]:The results are compared with the value defined for a random pattern, 2, with lower values suggesting a uniform pattern and larger values suggesting a clumped pattern (see [48], pp. 58-59 for details); and z is compared to a table of critical values for the standard normal distribution to detect significant departures from randomness [48]. The second procedure used for PCQM was suggested by Eberhardt [50] and analyzed further by Hines and O’Hara Hines, [47]:where s and are, respectively, the standard deviation and mean of the point to nearest individual distances. Critical values have been computed by Hines and O’Hara Hines (see [6], p. 210 for details). The expected value in a random population is 1.27, lower values suggest a regular pattern, and larger values indicate clumping.

Regarding quadrat counts, the data comes from  m quadrats. Morisita’s index of dispersion, , has been extensively used to evaluate the degree of dispersion/aggregation of spatial point patterns [51]. It is based on random or regular quadrat counts and is closely related to the simplest and oldest measures of spatial pattern, the variance to mean ratio [6, 30]. We used a standardized version () of the index which is robust to variation in sample number and size [51] and is obtained by first calculating the raw index:where is the number of samples and is the number of individuals per sample. The standardized Morisita index of dispersion () ranges from −1 to +1, with 95% confidence limits at +0.5 and −0.5. Random patterns give an of zero, clumped patterns above zero, and uniform patterns below zero (see [6], p. 217 for details). The second index used for quadrat counts is the variance to mean ratio, where a random pattern, described by the Poisson distribution, corresponds to the value 1:If the variance is larger than the mean, distribution is clumped; if it is smaller than the mean, distribution is regular [6]. To test significant departures from the random expectation, confidence envelopes using the test for degrees of freedom are calculated [30].

2.6. Statistical Analyses

To compare the different sampling methods, we used a resampling approach [52, 53]: (i) 25 sample points were randomly selected from the 30 available for each plot; (ii) the different estimators were used to calculate tree density; (iii) this procedure was repeated 100 times; (iv) a similar approach was followed to estimate tree density using quadrat counts, where 30 out of 36 were used at each iteration. This procedure allowed simulating the variability in tree density estimates obtained using the different methods. Using this simulated dataset of tree densities, we compared all the estimators using a generalized linear model (ANOVA). Standard contrasts were then used to compare tree densities obtained using quadrat counts (the benchmark method) with all the other methods/estimators (). In order to further compare the departure of each estimator from the benchmark method, we calculated the relative mean difference (for each of the six plots) between tree densities obtained by each estimator and the quadrat method and represented those values using a bar plot. In the same sense, the estimators were ranked, according to their distance to the benchmark method, and the ranks for the six plots were represented using a box-plot. To analyze estimator precision, we calculated the coefficient of variation (CV) of tree density for each estimator and divided it by the CV obtained using quadrats. Statistical analyses were all performed within the R software, version 3.2.3 (R Development Core Team, 2014) [54].

3. Results

3.1. Estimated Tree Densities

Tree densities ranged from 344 to 5056 trees/ha and . undulatum tree density ranged from 267 to 4233 tree/ha (Table 3). According to the general results, -Square sampling methods tended to underestimate the tree densities obtained with quadrats (Figure 3).

PlotTotal number of treesP. undulatum number (%)Total density (trees/ha)P. undulatum (trees/ha)


3.2. Comparison of Sampling Methods

The results of ANOVA, based on 100 simulations, revealed significant differences for the estimation of tree density among the different estimators: plot 1, and ; plot 2, and ; plot 3, and ; plot 4, and ; plot 5, and ; and plot 6, and . In a more detailed analysis, comparing each distance based estimator with the quadrat estimates, -Square and PCQM were consistently more accurate than -tree (Figures 46). -Square estimators , , and were much more accurate and precise than estimator that greatly overestimated tree density (Figures 47).

The PCQM estimators performed with mix results. In general and produced more accurate estimates than (Figure 4). Overall, the best estimators, in terms of originating a consistently low difference towards the benchmark, include and , (Figure 4). Also, the estimators that showed the lowest differences towards the benchmark, based on the results for the six plots, were to , , and (Figure 5). In the same sense, the estimators that scored the best approximations towards the benchmark more often, based on the six plots, were again to and (Figure 6).

Regarding precision, the more accurate estimators, mentioned above, were also among those with lower CV as compared to the values obtained with quadrat counts (Figure 7).

3.3. Spatial Pattern Analysis

Regarding -Square sampling, the test of significance for was above the critical value (1.96) for plot 2 only. For , according to critical values for this test statistic, plots 3 and 5 showed regular patterns, plots 1 and 4 showed aggregation, and plots 2 and 6 showed a random pattern (Table 4). Regarding PCQM, for plots 1 and 4, was above the critical value (1.96). For , according to critical values for this test statistic, plots 5 and 6 showed regular pattern, plots 1 and 4 an aggregated pattern, and plots 2 and 3 a random pattern (Table 4).


Plot 10.571.341.373.123.381.610.93−0.070.9432.87
Plot 20.622.
Plot 30.510.
Plot 40.581.551.312.982.971.391.18−0.491.4149.22
Plot 50.591.641.151.35−1.961.081.130.502.6693.04
Plot 60.550.961.281.39−1.831.071.070.501.7360.40

For quadrat counts, indicated random patterns for all plots, although with several values close to the limits of the 95% confidence interval (i.e., −0.5 to 0.5), for which already indicated a significant departure from randomness (Table 4).

4. Discussion

4.1. Tree Densities

The tree density values (trees/ha of P. undulatum) obtained in the present research are within the range of the results obtained for P. undulatum stands in Graciosa Island in a previous survey using PCQM [41]. This confirms the wide variation in tree density that can be found in the Azorean exotic woodland dominated by P. undulatum and will have implications in estimating and managing biomass production [38, 41]. Such density variations are dictated by the underlying physical environment and past disturbance history. The results also confirm the potential for the biomass valorization of this species, which together with other uses, such as honey and compost production [55, 56] and reforestation measures, could be included in a broader management program for this invasive plant.

4.2. Sampling Methods

In terms of accuracy of results, when compared with quadrat counts, the -Square and PCQM estimators provided the best results. -Square estimators evolved as methods to remove some bias due to nonrandomness associated with the nearest neighbor distance measurement [13]. The -Square estimators performed generally slightly better and globally more consistently than the other estimators. The usual -Square estimator, , showed a similar performance regarding and . However, in studies performed by Engeman et al. [21], performed slightly better than and . In our study appeared to be inaccurate and should therefore not be used in Azorean forests dominated by P. undulatum. The two compound estimators and suggested by Byth [14] and Diggle [20], respectively, were developed to create a more robust estimator using -Square distance measurements that would perform better in a variety of spatial patterns, relative to more simple estimators utilizing only one distance measurement [13]. Both estimators showed a similar performance in our study, while some authors argue in favor of [14]. -Square measurements are essentially nearest neighbor measurements modified only by restricting the direction in which measurements are made but have been shown to be accurate in many situations [29]. However, in our study -Square estimates tended to underestimate the results obtained with the quadrat method (used as a benchmark). This should be further analyzed and taken into account in future studies, although -Square sampling is simple to implement in the field, and this simplicity is often preferred by field workers.

Point Centered Quarter Method is often used in forestry, and we found that the estimator , given by Pollard [15], was among the most accurate among those tested. The advantage of this estimator is that it is very efficient in field sampling. Previous work [2, 3] concluded that PCQM was capable of yielding accurate results and was least susceptible to subjective bias than other methods [1]. However, it has the disadvantage of being very susceptible to bias whenever the spatial pattern is not random [15].

Our results showed that -tree method performed badly in all plots, showing a considerable overestimation of tree density. Klein and Vilkco [57] proposed two new estimators but it seems that these estimators still need more field work validation. Zobeiry [58] showed that -tree sampling techniques are not suitable for the natural forest of northern Iran because they consistently underestimated the true population parameters, a clumped pattern being the most reasonable cause of this behavior. However, -tree sampling performed well in others studies [1].

4.3. Spatial Pattern

In temperate forests, tree species can have random, aggregated, and even regular spatial patterns, to different degrees, and the spatial pattern of a given species may vary from place to place [33]. The major weakness of density estimators is that their bias is dependent on the spatial distribution of the population [10]. Some distance estimators have been shown to be unbiased over a wide range of spatial patterns if the estimators are adjusted according to the spatial pattern. However, this adjustment would need additional tests to determine a population’s spatial distribution before estimating density [10]. -Square sampling has been largely recommended for spatial pattern analysis based on both simulated patterns and mapped field data [26, 29, 31, 48]. However, the statistics used in our study showed somewhat irregular results. This might be explained if P. undulatum stands correspond to a spatial distribution with a slight deviation from randomness, that is, a small degree of aggregation. This was evident in the results obtained for the parameters used for spatial pattern analysis from quadrat counts, with several values very close to the limit for randomness in the standardized Morisita index, which already corresponded to a significant deviation from randomness when using the variance to mean ratio. Therefore, the studied plots are more likely to correspond to random or slightly aggregated distribution patterns. Among Azorean native trees, the aggregate pattern is predominant in saplings but adults are mostly randomly distributed [33]. Aggregate patterns of distribution are reported as the most commonly observed in nature [6, 59], caused by environmental heterogeneity [6] and nonlinearity of biological processes [60] and density-dependent mortality/self-thinning [33]. The main reasons leading to a clustered pattern in a population are the behavioral characteristics of the species and intra- and interspecific relationships [61]. Intraspecific aggregation should limit the mean species richness of the communities to a level less than what it would be if individuals were randomly distributed among the communities [59]. Spatial structure of forest is a complicated characteristic due to the complex factors influencing it [25]: climate, micro-site mosaic, regeneration methods, natural mortality of individuals, biological and ecological characteristics of organisms, natural disturbance (e.g., fire, wind, and landslides), and human activities.

Human activity is an important factor in managed stands, such as the case of P. undulatum that was widely cultivated throughout the world, including Atlantic (e.g., Azores, Jamaica) and Pacific islands (e.g., Hawaii) and South Africa [62]. It has become dominant in moist disturbed secondary forests and some primary forests from low to middle elevations [3639]. Perhaps the main effect of P. undulatum comes from its ability to penetrate natural ecosystems by the competitive exclusion of native species [63], regenerating readily and competitively in forest gaps [64] and having dense shade tolerant seedlings that outcompete native seedlings, which affects its distribution pattern [65]. Seedling and sapling aggregation may also result from the type of dispersion and seed rain [66]. Seed and seedling concentration may occur, for example, in places where animals frequently defecate [67] which is especially important since most Azorean tree species are dispersed by birds [68].

5. Conclusion

Distance methods also known as plotless sampling techniques were introduced because of the practical difficulties sometimes raised by quadrat sampling [69]. Plotless sampling is considerably more efficient than quadrat sampling, since searching and counting individuals in a large quadrat are very time consuming. However, accuracy and precision are important factors in determining an appropriate sampling method. This is because bias or imprecision in initial density estimates will propagate through derived metrics, such as basal area, biomass, and carbon storage. Among the tested methods, -Square sampling ranked more often as the closest estimator to the benchmark method. It is recommended for aggregated spatial patterns (a limitation for the use of PCQM). According to our results aggregation might occur in some of the stands. Moreover, -Square sampling provides more independent tree measures than PCQM (only 2 per sampling station, instead of 4), which might be useful when other parameters are also being sampled. Therefore, we suggest that -Square sampling might be used to estimate tree density in management programs dedicated to P. undulatum in the Azores.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The first author would like to acknowledge NATURALREASON, LDA, a company devoted to biofuels production, for supporting her research grant and for its financial contribution to the ongoing research project. This work is also funded by FEDER funds through the Operational Programme for Competitiveness Factors (COMPETE) and by National Funds through FCT (Foundation for Science and Technology) under the UID/BIA/50027/2013 and POCI-01-0145-FEDER-006821.


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