It is widely accepted that the structural complexity of a habitat can affect community-level properties like species diversity and richness [1] -[5] . Because of the wide-ranging consequences, it is important to be able to describe this structural complexity in quantifiable ways. This has proven to be a challenging endeavor, however, as familiar mathematical approaches (e.g., Euclidean geometry) have fallen short of providing accurate approximations of biological processes [6] [7] . This has prompted mathematicians and scientists to turn to lesser explored techniques. The product of one such exploration garnered much attention in the three decades since its inception: fractal geometry [6] -[10] .

While there are many definitions of varying mathematical accuracy for fractals [11] , Mandelbrot [8] [12] initially defined them as objects whose topological dimension is exceeded by their Hausdorff-Besicovitch dimension. For the purposes of this study, we will define a fractal as an object that displays some degree of self-simi- larity at any scale (for example, Figure 1). Voss ([13] ) provides a detailed overview of fractal geometry and the differences between it and Euclidean geometry. Briefly, initial applications for fractal geometry included the classification of coastlines, clouds, mountain range silhouettes, rivers, and a plethora of other natural features, as fractals (or, fractal-like). Fractal geometry has also been used to assess the spatial patterns of communities and forest in satellite images ([6] [11] [14] -[16] ); to explain allometric scaling and metabolic processes [17] -[20] ; and to examine the growth patterns of individual plants [21] .

The defining feature of fractals is that they are scale-independent (or, scale-invariant; [13] , see also [22] ). This property makes fractal geometry well suited to examine the structural complexity of various communities (e.g., coral reefs, grasslands, kelp forests), and even allows them to be compared to one another. Fractal dimension (D, the shape filling property of an object) has been coupled to the Shannon Index to decipher the relationship between the temporal and spatial complexity of an area [23] . This utility has led to many investigations of the spatial distribution of plants in ecological succession [8] [10] [23] -[27] . For example, in their survey of bird species in various habitats, MacArthur and MacArthur [1] found a correlation between foliage height and forest bird diversity. Their method of analyzing canopy structure, later expanded by MacArthur and Horn [28] , involves locating random vertical transects in plots and measuring where these points intercept foliage. Although this method appears to be valuable and flexible e.g. [29] , it is tedious and time consuming and thus not practical to implement in the field, especially with limited manpower.

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, [30] [31] ). These metrics only estimate individual-level parameters though, and do not represent community-level measurements. LAI, for one, calculates total leaf area per unit ground area, and provides a useful estimate of leaf surface area for further estimates of parameters such as photosynthesis and productivity measures [32] . This, however, is not an estimate of canopy structure, although higher values do indicate increasing complexity. Shipley and Lechowicz [31] recently introduced a measure for estimating leaf orientation across species, but suggest that metrics for canopy architecture are still needed. At scales smaller than that of forest canopies, there has been much interest in estimating the shape of structures such as leaves and branching patterns [17] [19] [20] [33] -[35] , as well as the effect of these shape patterns on dependent species [36] . Missing from these estimates is a rigorous and quantitative measure of canopy shape and associated gaps.

A major reason for the lack of any canopy shape metric is that there remains so much to understand theoretically regarding these characteristics [7] [37] , including how gaps influence recruitment e.g., [38] -[40] and affect associated understory species [41] [42] . Bollinger [43] investigated and reviewed the structure of communities at scales larger than patches and discussed changes in forest structural patterns over time. In terms of a rigorous quantification of these structural patterns, we must employ the unfamiliar approaches mentioned above. Here, we present a new method for quantifying forest canopy shape and gaps using fractal geometry, and show how this can be linked to changes in D over a vegetative chronosequence—a set of differently aged forests assumed to represent a realistic pattern of forest succession.

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 [9] [10] . Applying these techniques in the absence of digitized images (i.e., when using field measurements), however, is difficult. Past methods have included Mandelbrot’s perimeter-area method [12] , which is incongruous with an analysis taking into account not only length and width, but also height. An effective approach to applying to this three dimensional scenario is known as the box-count method (see Methods; see also [9] [13] [44] ). The result of this method is an estimate of D, which may become problematic when the smallest scale measured (here we used 0.25 m) approaches the size of individuals in the community. In this work, however, the communities measured contained trees greater than a fixed size (5 cm diameter at breast height, “DBH”) and, therefore, should be amenable to comparison across communities. The plot sizes used to analyze the chronosequences were chosen purposefully to allow for comparison of not only different communities, but communities at different stages of succession.

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; [9] [10] [13] ) requires another measure, complementary to fractal dimension. The manner in which an object fills space can be measured by the mean distance between points of the object, and this can differentiate objects of similar fractal dimension from one another. The two measurements taken together serve as unique identifiers of an individual plot, and allow for direct comparisons among plots of different stages of succession. We seek to obtain these measurements for a set of differently aged vegetative stands and show that our method can be readily applied in the field with minimal effort.

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 m² square plots in the SUNY Geneseo Roemer Arboretum (42˚47'16''N, 77˚49'25''W) and five 400 m² 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; Table 1). Plot sizes depended upon the tallest vegetation within an area to avoid plot sizes that would not capture the area studied; old field plots would have exaggeratedly low measurements of D when measured in a plot of height 20 m (see Figure 2(a)). Therefore, we used 8000 m³ plots for old growth forests, 1000 m³ plots for young growth forests, 125 m³ plots for shrubby habitat, and 27 m³ plots for grassland.

Although the structure of habitats has long been considered to have an important influence on niche availability, a direct quantification has been elusive [21] . The current work reports a simple technique that allows for the quantification of traditionally three-dimensional structures, such as forest canopies, using fractal geometry, as it has been developed for aquatic environments for over a decade [50] . We can estimate the structure of complex shapes using the box-count method to produce the fractal dimension for a variety of structures. The method described is relatively easy to implement and can be readily used to assess structural differences across systems; a tool that is sorely needed [7] . The 20 m² 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 [27] . We acknowledge, however, that we require further investigation to determine whether our method accurately estimates D for these plots—and thus they are in fact equivalent—or if our method is incapable of detecting real differences.

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 [21] . Given the complex effects that tree architecture can have on animal communities, it would be important to investigate how D scales up from individual trees to whole forest stands. Our results from the investigation into the shape of forest canopies suggest that broad canopy shapes for individual trees may influence larger canopy architecture (see also [27] ). Their work suggests that the structure of a single tree does scale up to describe the shape of the local neighborhood. Our method to efficiently calculate the fractal dimension of a vegetative stand may therefore be simplified even further; calculating D for a single tree may be sufficient to assess the structure of the surrounding habitat through time. This has profound implications for forest and wildlife management given the strong connection between landscape genetics and landscape ecology [51] .

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 [2] -[5] [36] . For example, Bélisle and Desrochers [52] have shown that birds clearly avoid open grasslands when possible; yet smaller animals may spend most of their lives in such habitats.

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 [53] .

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.