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A practical method for mapping trees using distance measurements.

INTRODUCTION

Accurate maps of the locations of trees are essential for spatially explicit studies of tree populations and forest dynamics (e.g., Sterner et al. 1986, Condit et al. 1992, Moeur 1993). However such maps are often difficult to obtain with traditional surveying methods, because the trees hinder line of sight measurements and the transit must be moved frequently to map an entire stand. An alternative method is to establish a grid of points with a transit, lay out measuring tapes on the ground at right angles between grid points, and then estimate tree coordinates from the tapes by eye or with a right-angled prism (e.g., Reed et al. 1989). Although GPS (global positioning system) devices can provide accurate locations of bench marks, current hand-held units are accurate only to [approximately]1-10 m and are adversely affected by a heavy canopy. In addition there is the problem of measuring the vector between the GPS receiver and the center of the tree.

In 1978 Rohlf and Archie proposed a method for mapping trees based entirely on tree diameter and tree-to-tree distance measurements. Distance measurements are preferable to angle measurements, because they are simpler and permit much higher accuracy to be obtained in the field. The triangulations required to calculate tree coordinates from the field data are easily performed by computer. The Rohlf-Archie method has been used in a number of published studies (e.g., Mitton and Grant 1980, Robertson 1984, Glitzenstein et al. 1986; with modifications in Kenkel 1988). However the technique as proposed is problematic for large numbers of trees, because it can be very difficult to isolate input errors (recording errors or gross measurement errors). In addition the computer program provided by Rohlf and Archie has been shown to generate large errors with both field collected and computer simulated data, raising the question of whether unacceptable accumulation of error is inherent to the technique (Hall 1991).

In 1990 we developed our own computer program (INTERPNT) for mapping trees using tree diameters and tree-to-tree distances, inspired by Rohlf and Archie's method (but without least squares optimization), and incorporating practical improvements from extensive field testing. These improvements included simplification of the information recorded in the field and the ability to isolate recording and gross measurement errors. The resulting "Interpoint method" was used to map trees in several long-term study plots at the Harvard Forest during the period 1990-1994. More recently the critical question of accumulation of error was resolved through extensive computer simulation of the technique. Our results suggest that the Interpoint method is both easy to use and highly accurate. In this paper we explain how the Interpoint method works, how the method was tested in the field, and how the method was tested by computer simulation. The INTERPNT program (with documentation and field instructions) is available on the Harvard Forest Web page.(4)

METHODS

Interpoint method

Overview. - The Interpoint method requires that the plot to be surveyed have at least three reference points (bench marks) with known Cartesian coordinates. Trees in the plot are numbered and the dbh (diameter at breast height) measured. Each new tree (target tree) is located by measuring the distances to three previously located trees (reference trees) or bench marks. A computer calculates the coordinates of the center of the target tree by triangulation and averaging, using the three triangles formed by the three distance measurements and the measured diameters at breast height. Possible errors are identified in the data analysis and investigated in the field.

Field measurements. - Field work for the Interpoint method includes establishing bench marks, labelling trees, measuring tree dbh, and measuring tree-to-tree distances, and proceeds as follows:

1) Three (or more) bench marks are located and permanently marked. Bench marks should be moderately spaced and their locations with respect to one another determined as accurately as possible. We found it convenient to use one corner of the plot as a bench mark, locating the other two bench marks at 15 m and 20 m along the two neighboring sides and carefully adjusting the points to create a 3-4-5 right triangle (i.e., 15-20-25 m).

2) Each tree to be mapped is labelled and measured for dbh to the nearest 0.1 cm. We used numbered aluminum tags at dbh height (1.37 m); the tags provided a convenient reference for setting the height of the measuring tape for the tree-to-tree measurements.

3) Beginning in the vicinity of the bench marks, and moving gradually across the plot, each target tree is located by measuring the horizontal distance to three reference trees (or bench marks). Reference trees close to the target tree (but not closer than 1 m) are selected. Pairs of reference trees that subtend an angle of [less than]20 [degrees] or [greater than]160 [degrees] at the target tree are avoided, because such pairs may magnify measurement errors (e.g., errors may increase by a factor of 20 at angles of 5 [degrees] or 175 [degrees]). Distances are measured from bark to bark to the nearest 0.01 m, holding the measuring tape as level as possible at dbh height on the target tree (a third person makes it easier to level the tape and a plumb bob makes it easier to measure distances to the bench marks). After each target tree is located it is flagged so that the surveyors can recognize it as a potential reference tree. Target trees may be recorded in any order; we found that recording them in order of tag number helped us ensure that all trees were accounted for. For each target tree the three reference trees are recorded in clockwise order as seen from above.

4) At the end of each field day, field measurements are entered into a computer and analyzed. Possible errors are noted and investigated at the beginning of the next field day.

Computer analysis. - Analysis of the field data by computer includes checking for errors and calculating Cartesian coordinates for all trees mapped to date, and proceeds as follows:

1) The list of located target trees is sorted so that each target tree is preceded in the list by the three reference trees or bench marks that were used to locate it. This procedure simplifies field operations by making it possible to record target trees in any order (if the computer assumes that target trees are listed in the order of actual measurement, then a single recording mistake could render a large data set virtually unusable). The sort also detects two critical errors that are difficult or impossible to detect by hand: (a) missing trees, i.e., references to trees that do not exist or have not yet been located; and (b) circular references, i.e., references (perhaps through a long series of trees) back to the target tree itself. These errors result from recording the wrong tree number or selecting a tree not yet located as a reference tree. Circular references are particularly difficult to isolate, since finding them may require examining all possible pathways from the target tree to trees in the last data set that was sorted successfully. Because the magnitude of such a search increases exponentially as the number of trees increases, it is important that new field data be processed regularly to avoid the possibility of an unreasonably long search.

2) Cartesian coordinates for the center of each target tree are calculated by triangulating from each of the three pairs of reference trees (or bench marks). Each pair of reference trees yields two possible locations for the target tree. The problem of mirror triangles is solved by recording the reference trees in clockwise order [ILLUSTRATION FOR FIGURE 1 OMITTED]. This procedure is easier and less prone to error than recording all three left/right pairs as F. J. Rohlf and J. W. Archie suggest.

3) In rare cases triangulation fails because of an open triangle (where the length of one side is greater than the sum of the lengths of the other two sides). Here the computer adjusts the measured distances from the target tree to the two reference trees to make the triangle close and permit triangulation to continue. The magnitude of the adjustment and the relevant tree numbers are noted and such cases are checked in the field for possible recording errors, measurement errors, or pairs of reference trees subtending angles [less than]20 [degrees] or [greater than] 160 [degrees].

4) For each target tree and associated reference tree (or bench mark) a "distance difference" is calculated as an indication of possible error. The distance difference is the calculated distance between trees (based on the calculated Cartesian coordinates) minus the measured distance (the measured bark to bark distance plus one-half the dbh of each tree). Target trees are marked in the computer output if the absolute value of any of the three distance differences is [greater than] 10 cm. Marked trees near the top of the sorted list are checked first in the field (often if a tree is marked then later trees measured back to it are also marked, so the first marked trees are investigated first).

5) Maps showing tree locations and tree numbers (or other collected data such as tree species) may be generated by computer using the calculated coordinates of the trees surveyed to date. Such maps are very useful for locating particular trees and investigating problems in the field as the survey continues.

6) Possible errors identified in the computer analysis (missing trees, circular references, open triangles, and marked trees) are all checked in the field. In our surveys, if a tree was marked but no recording or measurement errors were discovered, one or more of the reference trees were replaced with new reference trees (referred to below as "reference tree replacement"). In rare cases this was done more than once for the same target tree.

7) Overall accuracy may be improved by applying a "distance correction factor" to the tree-to-tree measurements in cases where there is independent and reliable evidence of measurement bias in a particular plot.

Field test

Harvard Forest plots. - The Interpoint method was used to map the locations of standing trees with dbh [greater than or equal to] 5 cm in seven rectangular plots at the Harvard Forest in central Massachusetts, USA (Table 1). The plots contained mature stands and ranged in size from 0.25 ha (200 trees) to 0.80 ha (889 trees). Average slopes ranged from level to 5 [degrees]. One plot (Plot 1) was dominated by Tsuga canadensis; the others by Quercus rubra and Acer rubrum. The T. canadensis plot, which has a small bog (10 m x 20 m) in the center of the plot, was mapped in the winter when it was possible to measure tree-to-tree distances directly across the ice of the bog. The other plots were mapped in the summer. Field measurements were performed by two or occasionally three people. Species, crown position, and health were recorded for each tree. Tree-to-tree distances were measured with fiberglass tape; measurements were not corrected for tape stretch.

Field check of accuracy. - Accumulation of error in one plot (Plot 3) was checked by directly measuring the distances between 18 pairs of widely spaced trees, including 10 pairs of trees -30 m apart and parallel to the short axis of the plot and 8 pairs of trees -75 m apart and parallel to the long axis of the plot. Tree pairs were selected so that no tree was closer than 5 m to the plot boundary, and the pairs were widely spaced in an effort to sample the entire plot. For each pair of trees, the dbh of each tree, the bark to bark distance from breast height on one tree to breast height [TABULAR DATA FOR TABLE 1 OMITTED] on the other tree, and the angle of the tape endpoints above or below horizonal were measured. Measured distances between trees were adjusted using the manufacturer's suggested correction for the stretch of the fiberglass tape under tension. These values were used to calculate the horizontal distance between tree centers. This measured distance was then compared to the calculated distance using the Cartesian coordinates from the Interpoint method.

Simulation test

Objectives. - Though the tree maps generated from our field data appeared to be quite accurate, our field test was not sufficient to quantify the accumulation of error or to rule out occasional catastrophic error. To address these issues we developed a computer simulation of the technique in which we attempted to model field conditions as closely as possible. Two measurement errors were simulated: (1) an elevation angle error associated with failure to hold the tape endpoints level (always positive), and (2) a linear measurement error associated with human error in reading the tape (positive or negative). The simulation permitted us to observe the build up of location error (defined as the distance between actual and calculated tree location) across a large plot for many hundreds of runs. The following questions were investigated with the simulation:

1) How much does location error accumulate across the plot? How much does location error vary as a function of the standard deviation (SD) of the linear measurement error? For a given standard deviation of linear measurement error, how much does location error vary between different runs?

2) Does reference tree replacement improve overall accuracy?

3) Does correction of the positive bias caused by elevation angle errors improve overall accuracy?

4) Does the shape of the plot have an effect on overall accuracy?

5) What is the effect of a single, large, uncorrected error?

6) What is the effect of a large measurement bias? Simulation of errors. - Tree-to-tree measurements include errors that always decrease measurements (negative bias; e.g., tape stretch), errors that always increase measurements (positive bias; e.g., catenary sag of tape), and errors that can be positive or negative (no bias; e.g., departure of tree cross sections from circular). Of these, the two errors judged to be largest and most difficult to correct for were modeled in the simulation:

1) The elevation angle error resulting from failure to hold the tape endpoints level. This error is always positive and is calculated as s = L(1/cos A - 1) where s = error in length, L = actual length, and A = tilt angle of tape from horizontal. In the simulation the tilt angle was represented by a normal distribution function (mean = 0) that provided both positive and negative angles (but note that s is always positive, creating a small positive bias). A standard deviation of 1 [degrees] was used in all cases except one. Experimentation in the field suggested that with care we could reliably hold the tape to within one-third degree of level by eye over a distance of five meters, so our results probably did not underestimate the actual elevation angle error.

2) The linear measurement error associated with human error in reading the tape. This error may be positive or negative. In the simulation it was represented by a normal distribution function (mean = 0) whose standard deviations were 1, 1.5, 3, 6, 12, or 24 cm. This wide range of standard deviations was chosen to exceed likely conditions in the field. Field data from the Harvard Forest plots suggested that the actual standard deviation of the linear measurement error in the Harvard Forest surveys was on the order of 1.5 to 3 cm. Note that in the simulation the average linear measurement error was at least an order of magnitude greater than the average elevation angle error.

Simulation of tree mapping. - Target trees were located at random in square or rectangular plots. Plots contained 1000 or 10000 trees, and tree density was set to 1000 trees/ha based on the Harvard Forest field data. The selection of reference trees was patterned as closely as possible after the field practice. Tree-to-tree measurements were simulated by adding the elevation angle error and the linear measurement error to the actual distance.

The practice of reference tree replacement was modeled in selected runs by calculating the distance difference for each target tree and its three reference trees. If the distance difference exceeded 3.5 times the standard deviation of the linear measurement error, then a new set of measurement errors was chosen by the usual process, simulating remeasurement in the field. If the new distance difference still exceeded 3.5 times the standard deviation of the linear measurement error, then all three reference trees (or as many as possible, but at least one) were replaced with new trees.

The elevation angle error always introduces a small positive bias in the tree-to-tree measurements. The possibility of correcting for this bias was modeled in selected runs by multiplying the simulated measured distances between trees by the cosine of the estimated average elevation angle of the tape.

The effects of plot shape on overall accuracy were tested by running the simulation for rectangular plots of different length-to-width ratios. The impacts of a single, large, uncorrected error were tested by assigning erroneous coordinates to one of the bench marks and examining location errors across the plot. The effects of a 0.25-1.0 cm measurement bias were tested by setting the mean linear measurement error to a range of positive and negative values.

Statistical methods. - Trees for 1000 tree runs, ordered on their distance from the origin, were divided into 10 bands of 100 consecutive trees each; while trees for 10000 tree runs were divided into 16 bands of 625 trees each. For each band, location errors were sorted into one of twenty equal classes to build a sample frequency distribution. In addition the sample mean, the sample standard deviation, and the maximum location error were calculated for each band.

The sample frequency distribution of location error for a single band typically has a bell shaped curve, somewhat skewed on the upper end. Since a large number of samples is used to calculate the sample mean and the maximum for each band, the sample means and maxima for each band over multiple runs should be nearly normally distributed. Each set of input conditions was tested with at least 100 runs, and the results were used to calculate sample frequency distributions of the mean and the maximum location error over all runs for each band.

The effects of various corrective methods (e.g., reference tree replacement) were evaluated by making at least 100 runs for each method. The means of the mean location error in the final band of trees were then compared statistically using a t test.

RESULTS

Field test

Harvard Forest plots. - The average measured distance (center to center) between target trees and reference trees for all seven plots was 4.96 m (Table 1). Measurement errors were detected by computer and corrected in the field for 2.1% of the target trees. Another 4.5% of the target trees showed distance differences [greater than]10 cm, but no errors were discovered in the field. Here one or more reference trees were replaced with new trees. At the end of the survey the maximum absolute value of the distance differences in all plots was 19 cm, and only 22 trees (0.67% of all trees) had values [greater than]10 cm. The average absolute value of the distance difference across all plots was 1.63 cm, while the average signed difference was close to zero (-0.03 cm).

The average time required across all seven plots to tag each tree, measure dbh, identify species, and estimate crown position and health was [approximately]6 d/ha. Of these tasks, only the first two were necessary to map the trees, and estimating crown position required the most time. The average time required to locate target trees, measure distances to three reference trees, flag located trees, analyze data, and make any necessary remeasurements was [approximately]9 d/ha. These estimates were based on two person crews, 8-h days, and a tree density of [approximately]1000 trees/ha. In most cases the crews were undergraduate students with little or no field experience.

Field check of accuracy. - A comparison of measured distances and calculated distances between widely spaced trees in Plot 3 showed close agreement (Table 2). For the 10 pairs of trees parallel to the short axis (50 m) of the plot, the average measured distance was 28.93 m, and the average absolute value of the distance differences was 4 cm. For the 8 pairs of trees parallel to the long axis (120 m) of the plot, the average measured distance was 76.08 m, and the average absolute value of the distance differences was 16 cm. Note that for each shorter pair the two trees were roughly equidistant from the bench marks, while for each longer pair one tree was much farther from the bench marks than the other tree.

In all cases but one, the calculated distance for the longer pairs was longer than the measured distance, suggesting (as expected) a tendency toward positive measurement bias and the accumulation of positive error with distance from the bench marks. Cartesian coordinates for the plot were recalculated using a distance correction factor of 0.9980, the ratio between the average measured distance between the longer pairs (76.08 m) and the average calculated distance between the longer pairs (76.23 m). Calculated distances using the new coordinates were then compared to the measured distances. For the 10 shorter pairs the average absolute value of the distance differences increased slightly from 4 to 6 cm, while for the 8 longer pairs it decreased significantly from 16 to 6 cm.

Simulation test

Accumulation of location error. - Results of 100 runs for 1000 and 10000 trees suggest that location [TABULAR DATA FOR TABLE 2 OMITTED] errors increase slowly as one moves across the plot, away from the initial bench marks [ILLUSTRATION FOR FIGURE 2 OMITTED]. As expected, location errors (measured here as the means of the mean location error) were roughly proportional to the standard deviation of the linear measurement error. In each case the slope of the curve was greatest in the first two or three bands and nearly constant thereafter, with a slight increase in the final band which may have resulted from the more limited choice of reference trees in the far corner of the plot. The slope in the middle of the curve could probably be extrapolated to furnish a good estimate of the location errors for plots with [greater than]10 000 trees.

The simulation showed moderate variation between runs but no evidence of catastrophic error (Table 3). Even for exceptionally poor measurements (SD of linear measurement error = 24 cm, SD of elevation angle error - 2 [degrees]), the mean of the mean location error in the final band of 10000 trees was only 1.23 m. The maxima of the maximum location error for a SP of linear measurement error of 3 cm were 32 cm and 51 cm for 1000 and 10 000 trees, respectively; note that these were the worst locations errors in 100 000 and 1 000 000 trees.

Reference tree replacement. - The value of reference tree replacement, which was used in mapping all of the Harvard Forest plots, was tested by comparing 100300 runs for each of the initial conditions described in Table 3, both with and without reference tree replacement. Results for the final band for both 1000 and 10000 trees showed that reference tree replacement improved the means of the mean location error in four of six cases, but worsened it in the other two cases. In all six cases the standard deviation of the mean location error was improved. The results suggested a general trend toward improvement; however no cases were statistically significant at the 0.05 confidence level.

Elevation angle bias correction. - The value of correcting for elevation angle bias was tested by comparing location errors in sets of 100 runs with an angle error of 1 [degrees] and angle bias corrections of 0.0 (no correction) to 2.5 [degrees] in 0.5 [degrees] increments [ILLUSTRATION FOR FIGURE 3 OMITTED]. Results showed that the location error in the final band decreased as the angle correction increased from no correction to 1.0 [degrees], but then increased beyond the uncorrected value as the angle correction increased from 1.0 [degrees] to 2.5 [degrees]. A t test showed that the reduced mean of the [TABULAR DATA FOR TABLE 3 OMITTED] mean location error for the 1.0 [degrees] correction group was better than no correction at a significance level of 0.02; while the 2.5 [degrees] correction was worse than no correction at a significance level of 0.00. Elevation angle bias correction thus appears to be effective if there is a reliable estimate of the actual angle error for a given plot; otherwise there is a significant risk of overcompensating and decreasing overall accuracy.

Shape of the plot. - The effect of the shape of the plot on location error was tested by comparing location errors for 100 runs for rectangular plots of the same area but different length-to-width ratios (Table 4). As expected, the mean and standard deviation of the mean location error in the final band increased as the length-to-width ratio increased. In a long, narrow plot there is a greater distance (and thus a greater number of trees) between the initial bench marks and the far end of the plot; also there may be some loss of accuracy caused by a greater number of trees next to the plot boundary, where the choice of reference trees is more limited.

Effect of a large error. - The effect of a single, large, uncorrected error in a single run was tested by introducing an initial bench mark error in a plot of 160 trees and examining the results for each band of 10 trees [ILLUSTRATION FOR FIGURE 4 OMITTED]. The distance differences and location errors, though initially large, decreased rapidly and by the final band were comparable to a similar run made with all bench marks correctly located. These results suggest that the averaging process of the Interpoint method (using three triangles to locate each target tree) is quite effective in reducing the propagation of a large error.

Measurement bias. - The effects of a systematic measurement bias were tested by comparing location errors for different values of the mean linear measurement error [ILLUSTRATION FOR FIGURE 5 OMITTED]. Results of 100 runs for 1000 and 10 000 trees showed that the location error was directly proportional to the mean linear measurement error for larger values of the mean. However local error, measured as the difference between the calculated distance between target tree and reference tree minus the (simulated) measured distance, remained unchanged. Thus a large measurement bias has a significant impact on the accumulation of location error across the plot, but little impact on the accuracy of a tree's calculated position relative to its immediate neighbors. Note that actual measurement bias should be well under 1 cm in most cases. For example, over a distance of 5 m, a catenary sag of the tape of 2.5 cm causes an error of +0.03 cm, an elevation angle error of 1 [degree] causes an error of +0.08 cm, stretch of a steel tape at 2 kg tension causes an error of -0.01 cm, stretch of a fiberglass tape at 2 kg tension causes an error of -0.10 cm, and a temperature change of 10 [degrees] C causes a steel tape to expand or contract by 0.06 cm.
TABLE 4. Computer simulation of the Interpoint method for 1000 trees
in rectangular plots of the same area but different length-to-width
ratios.

Plot       Mean     SD      Max.     Mean      SD      Max.
length/   of the  of the   of the   of the   of the   of the
width     means   means     means   maxima   maxima   maxima
ratio      (m)     (m)       (m)      (m)      (m)     (m)

1         0.088   0.037     0.231    0.172    0.048    0.320
2         0.121   0.054     0.281    0.213    0.062    0.369
3         0.122   0.057     0.315    0.211    0.064    0.387
4         0.130   0.063     0.343    0.222    0.070    0.434

Notes: Each row shows the distribution of the mean and the maximum
location error in the final band (trees 901-1000) for 100 runs of
the simulation model. Standard deviation of linear measurement
error = 3 cm, and standard deviation of elevation angle error = 1
[degrees].


DISCUSSION

Any evaluation of surveying methods must consider accuracy, speed, equipment, and personnel. After extensive field testing we found that the Interpoint method is easy to use, requires no special equipment or training, results in a computerized data set, and virtually eliminates gross errors. Mapping can be done while the trees are in leaf, because there is no need for extended line of sight measurements. The simulations showed that the technique is highly accurate for plots of moderate size if reasonable care is taken in setting out the bench marks and measuring tree-to-tree distances. The technique is best at estimating the relative locations of neighboring trees, an advantage for neighborhood studies. We found no evidence for the catastrophic accumulation of error reported by Hall (1991), even when measurement accuracy is quite poor. On the contrary, the averaging process appears to reduce the propagation of large, individual errors.

Drawbacks of the Interpoint method include:

1) Field data must be analyzed and problems checked on a regular (preferably daily) basis.

2) When a target tree is marked and no immediate errors are discovered, backtracking to search for errors among previous trees is time consuming and often impractical. Our field experience suggests that reference tree replacement solves this problem in nearly all cases, without the need for backtracking. We suspect that reference tree replacement helps to contain the effects of previous minor errors, though the results of the simulation on this point were not conclusive.

3) Measurement bias must be kept to a minimum to control accumulation of error, particularly in large plots. An independent check of accuracy is recommended for plots [greater than]100 trees, e.g., through a series of long distance tree-to-tree or tree-to-bench mark measurements. Measurement bias, if discovered, can be corrected for to improve overall accuracy. For plots [greater than]1000 trees, accumulation of error can be controlled by combining the Interpoint method with traditional methods; e.g., by using a transit to divide the plot into subplots and using the Interpoint technique to map trees in each subplot.

4) Tree-to-tree distance measurements may be difficult or impractical in steep or rough terrain.

Different surveying methods provide different compromises between time and accuracy. For example, the times reported by Reed et al. (1989) for mapping trees with the prism technique (1 d/ha for three people) are significantly shorter than the times reported here (9 d/ha for two people), though Reed's estimate assumed an experienced crew and did not include the time required for a surveyor to set out the original 50-m grid. On the other hand the Interpoint method appears to be significantly more accurate than the prism technique; the average difference in coordinates for trees remeasured by Reed et al. (1989) in 18 50 x 50 m plots was 16 cm (corresponding to a Euclidean distance of [approximately]23 cm), while our simulation of the Interpoint method predicts an average location error of [approximately]5.5 cm (for a 50 x 50 m plot, SD of linear measurement error = 3 cm, sp of elevation angle error = 1 [degree]) with even better accuracy on a local scale.

The Interpoint method can be extended in a variety of ways. New technology for measuring distances (e.g., laser or ultrasonic range finders) may be faster than measuring tapes, though such devices should be tested for accuracy and especially for measurement bias before being used with this technique. Stationary objects other than trees (e.g., herbs or shrubs) can be mapped by measuring distances between plants or between plants and grid points. And the method can be extended to map objects in three dimensions, as Rohlf and Archie (1978) suggest, by selecting four bench marks with known three-dimensional coordinates, and for each target object measuring distances to four reference objects.

ACKNOWLEDGMENTS

The authors thank T. Allison, C. Canham, D. Foster, J. Glitzenstein, D. Katz, and A. Lewis for many helpful suggestions, and M. Fluet, J. Gerwin, M. Kennon, K. LeClaire, R. Lent, P. Micks, K. Newkirk, E. Nilson, T Peterson, J. Quisel, A. Smyth, and T. Zebryk for assistance with the field work. The research was supported by the National Science Foundation and is a contribution from the Harvard Forest Long-Term Ecological Research Program.

4 URL = http://lternet.edu/hfr

LITERATURE CITED

Condit, R., S. P. Hubbell, and R. B. Foster. 1992. Short-term dynamics of a neotropical forest: change within limits. Bioscience 42:822-828.

Glitzenstein, J. S., P. A. Harcombe, and D. R. Streng. 1986. Disturbance, succession, and maintenance of species diversity in an east Texas forest. Ecological Monographs 56: 243-258.

Hall, R. B. W. 1991. A re-examination of the use of interpoint distances and least squares in mapping forest trees. Ecology 72:2286-2289.

Kenkel, N. C. 1988. Pattern of self-thinning in jack pine: testing the random mortality hypothesis. Ecology 69:10171024.

Mitton, J. B., and M. C. Grant. 1980. Observations on the ecology and evolution of quaking aspen, Populus tremuloides, in the Colorado front range. American Journal of Botany 67:202-209.

Moeur, M. 1993. Characterizing spatial patterns of trees using stem-mapped data. Forest Science 39:756-775.

Reed, D. D., H. O. Liechty, and A. J. Burton. 1989. A simple procedure for mapping tree locations in forest stands. Forest Science 35:657-662.

Robertson, J. G. M. 1984. Acoustic spacing by breeding males of Uperoleia rugosa (Anura: Leptodactylidae). Zeitschrift fur Teirpsychologie 64:283-297.

Rohlf, F. J., and J. W. Archie. 1978. Least-squares mapping using interpoint distances. Ecology 59:126-132.

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Author:Boose, Emery R.; Boose, Emery F.; Lezberg, Ann L.
Publication:Ecology
Date:Apr 1, 1998
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