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There
is a well known connection between the structural complexity of vegetative
stands and ecosystem properties. Developing methods to quantify this structural
complexity is an important goal for ecologists. We present an efficient and
easily implemented field technique for calculating the shape of forest
canopies, and the shape of forest stands as succession occurs, using fractal
geometry. Fractal geometry can be used to describe complex, non-Euclidean
objects that are common in natural systems. We tested the use of this tool in
22 vegetative and forested plots in Western New York State, USA. We found an
asymptotic relationship for fractal dimension (D) as a function of basal area (BA;
r^{2}= 0.68). In a randomization test to investigate the robustness of D to different tree canopy shapes, we
found that D was sensitive to canopy
shape switching, suggesting that the method is able to differentiate among
similar forests comprised of species having different shaped crowns. We
conclude that the shape is conserved in vegetative areas as they progress from
one stage of succession to the next (range of mean D: 2.56 to 2.68 across stages). Furthermore, we conclude that the
shape filling properties—i.e.,
distribution of trunks and limbs in a forested area, measured as mean
distance—are also conserved across vegetational chronosequences (F = 1.3189, df
= 8, 3, p = 0.3341).

It is widely accepted that the structural complexity of a habitat can affect community-level properties like species diversity and richness [

While there are many definitions of varying mathematical accuracy for fractals [

The defining feature of fractals is that they are scale-independent (or, scale-invariant; [

Other methods for estimating and quantifying aspects of canopy structure include leaf area index (LAI), leaf mass per unit area (LMA), and specific leaf mass (SLM, [

A fern frond is an example of a fractal object. Note how individual branches resemble the larger fern as a whole

A major reason for the lack of any canopy shape metric is that there remains so much to understand theoretically regarding these characteristics [

While this is a novel approach in our system, techniques for determining the fractal dimension of surfaces and digital images have been practiced for years [

Despite the high degree of simplification required, approximating a treetop as a simple Euclidean shape works well on a larger scale. This should not be confused with the scale-independent nature of fractals; a fractal at any scale is defined as being self similar, but defining what part of an object has fractal-like behavior does require a scalar dependence. Scalar dependence greatly simplifies using the box-count method to analyze an entire region of vegetation. However, determining the fractal dimension of a region does not produce definitive measures of comparison; a young-growth forest and a meadow may have the same fractal dimension while occupying physical space in very different ways. To account for the gaps that most natural objects have (referred to having lacunarity; [

We split our study into two parts: first, we determined if there is a link between D and basal area per hectare (BA) as a proxy for quantifying forest canopy shape; and, second, we determined if there is a noticeable vegetation shape change during succession. First, we established five 400 m^{2} square plots in the SUNY Geneseo Roemer Arboretum (42˚47'16''N, 77˚49'25''W) and five 400 m^{2} square plots in the SUNY Geneseo/Genesee Valley Conservancy Research Reserve (42˚49'59''N, 77˚48'13''W). Second, to address the relationship between D and successional stage, we used three plots per stage, located in different areas of each of the above locations and an additional site along the southwest shore of nearby Hemlock Lake (approximately 42˚N 68'19''N, 77˚60'95''W; ^{3} plots for old growth forests, 1000 m^{3} plots for young growth forests, 125 m^{3} plots for shrubby habitat, and 27 m^{3} plots for grassland.

In each plot we calculated D and total basal area per hectare (BA). To calculate D for each plot we recorded x-y coordinates, diameter of the tree trunk at breast height (DBH; ~1.37 m), height to lowest live branch (HLLB), and average canopy radius for each tree with a DBH ≥ 5 cm. We determined heights using a laser range finder to determine distance to the trunk and a clinometer to determine angles (see

Field measurements required to estimate the fractal dimension (D) of a forest plot. (a) Example of a sample forest plot broken down into small cubes. (b) Measurements of individual trees involve estimating distance to tree using a laser range finder and estimating angles to the bottom of the tree (theta 1), lowest live branch (theta 2), and top of the tree (theta 3). The average canopy radius is r. These measurements allow for the calculation of height to the lowest live branch and tree top. These data together allow for the reconstruction of forest canopies in a three- dimensional matrix

. Plot size at each sampling area based on the stage of sccession represented

Successional Stage | Sampling area | ||||||
---|---|---|---|---|---|---|---|

Roemer Arboretum | Research Reserve | Hemlock Lake | |||||

Plots | Size (m^{2}) | Plots | Size (m^{2}) | Plots | Size (m^{2}) | ||

Old Field | 2 (A, B) | (A)1.22 (B) 0.91^{} | 1 | 2.13^{} | 0 | N/A | |

Shrubland | 2 (A, B) | (A) 1.52 (B) 2.13^{} | 1 | 1.83^{} | 0 | N/A | |

Young Growth | 1 | 5.18^{} | 2 (A, B) | (A) 12.49 (B) 9.14^{} | 0 | N/A | |

Old Growth | 0 | N/A | 0 | N/A | 3 (A, B, C) | (A) 24.38 (B) 27.43 (C) 25.3^{} | |

D was calculated for plots using the box-count method (see [^{3}) cells, each with dimension 0.25 m on a side. We placed values of “1” in cells that were intercepted by the tree canopy or trunk (otherwise cells were set to “0”). The program counts the number of “1’s” for each box size, in increments that double the length of each side (i.e., sides of length 0.25, 0.5, 1, 2, 4, 8, and 16 m). D is estimated by regressing the log of the number of boxes intercepted by trees against the log of the reciprocal of the length of one side of individual cells [

The relationship between D and BA was assumed to fit the equation

where D is the fractal dimension, a and b are fitted coefficients, e is the base of natural logarithm, and BA is the basal area. We use this two-parameter asymptotic function that assumes the relationship goes through the origin since a community without vegetation (BA = 0) would have a fractal dimension D = 0. Flat sheets of continuous vegetation (e.g., a grassland measured at a large scale) would have D approaching 2, but since we are investigating the shape of tree canopies, the absence of trees would not have this estimate of shape. Since an object with D = 3 represents a completely filled cube, the relationship was assumed to rise asymptotically with the constraint that D ≤ 3.0. Interestingly, forests may have D < 2.0 when their canopies contain relatively few trees with non-contiguous canopies. We fit this relationship using SPSS ver. 10.0 [

To assess the sensitivity of the method to the variability in the shape of individual tree crowns, we conducted randomization tests using one of the mature forest plots. Existing tree crown shapes were assigned randomly within each plot while all other tree parameters remained constant (i.e., the x-y coordinates, HLLB, canopy radius, and tree height were not changed). Fifty canopies were created and D estimated for each plot. We then created a forest with all ellipsoids, calculated D, and tested this against this distribution of D’s in the randomized canopies using a one-sample t-test. This was then repeated for upward pointing crowns and cylinders. Normality tests for the distributions of D were done using the Kolomogorov-Smirnov test.

The plots in each location were used to estimate the D and mean distance of the chronosequences (

We measured the height of the tallest plant in the chosen plot, x-y coordinates, HLLB and DBH (defined above), and the average canopy radius. We did not assign trees in plots with dense tree growth their own Euclidean shape for leafy areas because we could not readily determine the boundaries for each tree in this situation; rather, the whole area above the HLLB was treated as a solid. To account for gaps in these Euclidean approximations, each tree trunk was treated as a cylinder, filling space below the HLLB, and serving as a proxy for empty space above the HLLB.

To avoid the overly tedious work of finding coordinates of each blade of grass in grassland plots, an average height for the plot was taken as the grassland plots included here had patches that were noticeably taller than other patches. Shrubland plots had no HLLB and thus were given Euclidean shapes starting at ground level, with coordinates located at the center of the shrub. We used the program outlined above to calculate D for each plot. For this investigation, however, we used a variation of this model to calculate the mean distance of each plot. The program calculated the average distance between all pairs of occupied cells (not merely the distance between points on the surface of different parts of the target object). That is, for each cell with a “1” (indicating the presence of a stem or branch), the program calculates the distance (in number of cells) to every other cell in the plot that has been assigned a value of “1”. Thus, our measure of mean distance serves as a reasonable estimate of the lacunarity of the plot. We preformed an analysis of variance on the outputs of the D and mean distance assessments in R v. 2.15.0 [

Our results indicate that the complexity of the forest canopies, as estimated by D, increased asymptotically with the basal area of the forest plots (D = 2.2885 × (1 − e^{−1.1173BA}, r^{2} = 0.68) with a 95% confidence interval about the asymptote (2.2885) of 0.135 (^{2} values for the regressions used to determine D exceeded 0.98 for all determinations of D.

Furthermore, the randomization tests revealed that, for this mixed-shaped forest canopy dominated by ellipsoids, changing to all ellipsoids did not significantly alter D (t = 1.64, df = 49, p = 0.11). Changing the canopy to all cones reduced D significantly (t = 57.5, df = 49, p < 0.001) while cylinders increased D (t = 29.3, df = 49, p < 0.001) (

Finally, we found no statistical difference between the average fractal dimensions of the vegetative areas repre- senting discrete successional stages (F = 0.213, df = 8, 3, p = 0.8846;

Relationship between fractal dimension and basal area of plots. The line was fit using non- linear regression

Results from the randomization test. Canopy shapes were chosen randomly and replaced existing trees. All other factors remained unchanged (e.g., x-y coordinates, height, etc.). Only the forest with canopies fixed to all ellipsoids was not significantly different (one-sample t-test) from the dis- tribution of 50 randomizations of a forest plot with randomly selected canopy shapes. When canopies were fixed as cones and cylinders the estimate of D was signifcantly less than and greater than the randomized forest plots, respectively

Fractal dimensions for each community type were not statistically different, suggesting conservation of overall shape, as quantified by the fractal dimension, during succession (F = 0.213, df = 3, 8, p = 0.8846)

Mean distances for each community type were not statistically different, sugesting conservation of shape filling properties during succession (F = 1.3189, df = 3, 8, p = 0.3341)

Although the structure of habitats has long been considered to have an important influence on niche availability, a direct quantification has been elusive [^{2} plots each took between 30 minutes to an hour for one individual to complete. Teams of two researchers would be able to complete a dozen or more plots per day. This compares favorably against other techniques that are relatively tedious in their measurements [

Our results suggest that shape and structure are conserved properties of vegetative stands throughout the process of succession. It is important to note that although shape (D) changes with BA, our results suggest that forest development is more complex than simply “trees are getting bigger”. We have shown that gross morphology of the forest changes nonlinearly with BA. This change and the functional relationship between BA and D are likely to be important influences on species using the forest canopy (e.g., arthropods, or birds). The results from the randomization test suggest that switching canopies from the observed shapes to canopies with all cones or cylinders had a large statistical effect. This suggests that for these eastern deciduous forests the shape of individual trees influences the overall structure of the forest canopy, an effect that this method clearly detects.

Furthermore, combining the descriptive but insufficiently unique metric of D with an estimate of the mean distance of points within an object yielded novel results about the spatial patterns of vegetative stands. We expected that stands at different stages of succession would exhibit significantly different measures of D and mean distance; our results, however contradict this expectation on both fronts (at least, in the type of environment included in our study). First, similar values for D values across successional stage may be attributable to genetic processes occurring at the individual level, as it has been shown to be heritable in trees [

In addition, the mean distance assessment showed no differences among the four stages of succession we considered here. This finding seemed counterintuitive as we anticipated that in the unlikely event that plots in different successional stages exhibited equivalent D, they would likely differ in the way they occupied the space, i.e., in their lacunarity. For instance, a purely grassy plot would tend to have a uniform height and consist of many stalks or blades of plants, while a mostly shrubby plot would consist of several bushes with extensive branching—more lateral structures than would be expected in a grassland and potentially greater distance between structures. This might then contribute to the expected differing degrees of animal species richness and diversity in plots displaying different stages of succession. Instead, we can exclude this feature as a possible cause and conclude that, while D for a vegetative stand may not differ greatly during the course of succession, the functional D may change. That is, the animals that interact with that vegetative stand may experience different aspects of its shape and structure and therefore preferentially move through and settle in one stand over another, a conclusion that has considerable support [

The results we present here offer more encouraging signs that the use of fractal geometry in ecology can yield critical insights into natural phenomena and processes. We acknowledge, however, that our sample size is small, and our geographic focus is restricted. Our results are thus constrained and may be applied only to northeastern deciduous communities. Nevertheless, these results still may help forest managers manipulate canopy structures for certain canopy-dwelling species. Our results also indicate that selective logging could be accomplished to maintain (or alter, as necessary) the structure of a forest. How the shape of a forest canopy influences community properties, such as animal communities, plant species composition, and ecosystem services, remains an important research priority. The method we present here is useful for rapid assessment of community structure, relying on the determination of the fractal dimension D. It is our hope that the expanded use of this method will lead to a better understanding of the human-induced impact on the environment [

We thank C. Leary, D. Ruppe, S. Vrooman, H. Miller, N. Sprentall, and J. Varughese for assistance with this work, as well as helpful feedback on early versions of this manuscript.