Corresponding author: Carmelo Maria Musarella ( carmelo.musarella@unirc.it ) Academic editor: Peter de Lange
© 2018 Carmelo Maria Musarella, Ana CanoOrtiz, José Carlos Piñar Fuentes, Juan NavasUreña, Carlos José Pinto Gomes, Ricardo QuintoCanas, Eusebio Cano, Giovanni Spampinato.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Musarella CM, CanoOrtiz A, Pinar Fuentes JC, NavasUrena J, Pinto Gomes CJ, QuintoCanas R, Cano E, Spampinato G (2018) Similarity analysis between species of the genus Quercus L. (Fagaceae) in southern Italy based on the fractal dimension. PhytoKeys 113: 7995. https://doi.org/10.3897/phytokeys.113.30330

The fractal dimension (FD) is calculated for seven species of the genus Quercus L. in Calabria region (southern Italy), five of which have a marcescentdeciduous and two a sclerophyllous character. The fractal analysis applied to the leaves reveals different FD values for the two groups. The difference between the means and medians is very small in the case of the marcescentdeciduous group and very large when these differences are established between both groups: all this highlights the distance between the two groups in terms of similarity. Specifically, Q. crenata, which is hybridogenic in origin and whose parental species are Q. cerris and Q. suber, is more closely related to Q. cerris than to Q. suber, as also expressed in the molecular analysis. We consider that, in combination with other morphological, physiological and genetic parameters, the fractal dimension is a useful tool for studying similarities amongst species.
deciduous, dimension, fractal analysis, phenotype, sclerophyllous, species, Calabria
Quercus L. is an important genus containing several species of trees dominating different forest communities. The ecological and economic role of Quercus spp. is well known (
In the genus Quercus have been counted between 300 (
Leaf morphology has been studied throughout the history of botany, using leaf shape, edge, vein arrangement, hairiness and other features as important characters in systematics (
Numerous authors have noted the comparative inaccuracy of early descriptive and biometric studies (
In their study of several Quercus species,
We calculated the fractal dimension by the boxcounting method integrated in the ImageJ software (
The main aim of this work is to establish an analysis of similarity of leaf shape amongst seven species in the genus Quercus from Italy and corroborate our previous studies (
In this work, we analysed 7 species living in Calabria using 275 tree samples belonging to Quercus robur subsp. brutia, Q. cerris, Q. congesta, Q. crenata, Q. ilex subsp. ilex, Q. suber and Q. virgiliana. Orientation largely determines the amount of light the leaves receive for photosynthesis and their size can thus be affected by this greater or lesser exposure to light. For this reason, samples were taken from the four cardinal points on each tree to examine the possible influence of orientation on leaf development. A total of 1,099 leaves were analysed from 120 samples of Q. robur subsp. brutia, 120 from Q. cerris, 154 from Q. congesta, 147 from Q. crenata, 240 from Q. ilex subsp. ilex, 139 from Q. suber and 179 from Q. virgiliana. All the leaves were colourscanned in a scanner with a resolution of 1200 dpi and 24bit colour. After scanning, the leaf was transformed to image 8bit greyscales and the image was segmented by selecting the greyscale between 111 and 126. We opened this image with the ImageJ programme in order to determine its fractal dimension (FD).
Fractal geometry is the most suitable method for characterising the complexity of the vascular system or other mathematically similar structures such as stream drainage networks in chicken embryos or the distribution of the vascular system of a leaf (
All manmade objects can be described in simple shapes using Euclidean geometry. However, natural objects have irregular forms that cannot always be represented using this method (
Due to the recentness of the discovery and its wide range of applications, there is still no universal definition of what actually constitutes a fractal. They are thus described according to their common properties: specifically, they must have the same appearance at any scale of observation, meaning that a fractal object can be broken down into parts, each of which is identical to the whole object (selfaffinity or selfsimilarity); they must have a fractional and not a whole dimension (fractal dimension); and finally the relationship between two of their variables must be a power law (where the exponent is its fractal dimension,
When an object is totally selfsimilar, such as the mathematical fractal known by the name of the Koch curve (Figure
A unit segment can be divided – for example – into three pieces similar to the original, each with a length of 1/3. In general, where N(h) is the number of pieces with a length h, it follows that N(h) ∙ h^{1} = 1. If we now look at a square with a unit side, we can break it down into 9 = 3^{2} smaller squares with a side of ⅓; that is to say N(h) ∙ h^{2} = 1. Finally, in the case of a cube, it is easy to see that the following is true: N(h) ∙ h^{3} = 1. That is, the exponent of h coincides with the topological and Euclidean dimension of the straight line (1), the square (2) and the cube (3) (
By extrapolation from this concept, if the object is completely selfsimilar, there is a relationship between the scale factor h and the number of pieces N(h) into which the object can be divided, which is given by N(h) = (1/h)^{D}; that is to say
Thus the fractal dimension of the Koch curve is:
a number that is very similar to the FD of the English coastline.
However, natural objects like leaves are not perfect fractals, as they are not totally selfsimilar but are said to be statistically similar. In this case, the value of their fractal dimension is known by the name of HausdorffBesicovitch and is:
The calculation of this limit is somewhat complicated and requires the use of different algorithms such as dilation methods, the perimeter method, Grassberger and Procaccia’s correlation dimension and boxcounting method. This last is the most widely used as it is very simple to implement with computer technology and highly accurate (
To find the fractal dimension of a digital image using the boxcounting method (
Number of boxes occupied for each box size.
Label  C2  C3  C4  C6  C8  C12  C16  C32  D 
QCONGESTA1_E_01  358874  166858  97125  44308  25268  11452  6553  1727  1.93 
The graphic representation of the regression line and the point cluster shows two very clearly differentiated parts. The minimum and maximum box size is therefore very important when applying this method. In fact, the approximation error must be reduced by selecting points with a “more linear” form as a box size.
The FD was calculated by the boxcounting method (
The boxcounting algorithm was then applied to this blackandwhite image of the venation network of the leaf to calculate the FD with box sizes (h) ranging from 2 to 32. Specifically, the image is covered with a grid of squares initially with side 2 and subsequently with squares with sides 3, 4, 6, 8, 12, 16 and 32 (in the image C2, C3, C4, C6, C8, C12, C16 and C32). Table
Once the points were represented (log(1/h), log(N(h)), we calculated the regression line (Figure
For the statistical treatment, the mean FDs were obtained for each species and an analysis of variance was undertaken to test for significant differences amongst the means. First, the ShapiroWilk normality test and the difference between the mean, median and kurtosis indicate that our data do not follow a normal distribution (Table
Descriptive statistics of FD values for each species and orientation.
Taxa  Median  Mean  Variance (n1)  Kurtosis (Pearson)  St. root of the variance  St. root [kurtosis (Fisher)]  

North  Q. robur subsp. brutia  1.5440  1.5290  0.0730  1.3300  0.0192  0.8327 
Q. cerris  1.6760  1.6676  0.0375  0.5768  0.0098  0.8327  
Q. congesta  1.8780  1.8310  0.0138  1.8836  0.0032  0.7587  
Q. crenata  1.9195  1.8669  0.0172  6.2735  0.0040  0.7497  
Q. ilex subsp. ilex  1.3530  1.3804  0.0297  0.9245  0.0055  0.6133  
Q. suber  0.8620  0.9001  0.0703  0.3360  0.0173  0.7879  
Q. virgiliana  1.9310  1.9192  0.0016  7.4011  0.0003  0.6876  
South  Q. robur subsp. brutia  1.7675  1.6220  0.0895  1.6597  0.0235  0.8327 
Q. cerris  1.6600  1.6190  0.0337  0.3597  0.0089  0.8327  
Q. congesta  1.9000  1.8749  0.0058  2.8406  0.0014  0.7587  
Q. crenata  1.9200  1.8803  0.0106  2.8957  0.0025  0.7497  
Q. ilex subsp. ilex  1.3610  1.3442  0.0149  0.2129  0.0028  0.6133  
Q. suber  0.9395  0.9487  0.0408  0.0321  0.0100  0.7879  
Q. virgiliana  1.9120  1.8780  0.0060  1.1207  0.0013  0.6876  
East  Q. robur subsp. brutia  1.8405  1.7336  0.0428  0.2321  0.0112  0.8327 
Q. cerris  1.8360  1.8110  0.0143  0.0039  0.0037  0.8327  
Q. congesta  1.9230  1.9215  0.0008  2.2392  0.0002  0.7587  
Q. crenata  1.9270  1.8476  0.0257  1.2883  0.0060  0.7497  
Q. ilex subsp. ilex  1.3170  1.2954  0.0196  1.7224  0.0036  0.6133  
Q. suber  0.8850  0.9059  0.0475  0.3256  0.0117  0.7879  
Q. virgiliana  1.9445  1.9287  0.0032  11.3639  0.0007  0.6876  
West  Q. robur subsp. brutia  1.5715  1.5676  0.0800  1.2799  0.0210  0.8327 
Q. cerris  1.6050  1.6116  0.0643  2.0300  0.0169  0.8327  
Q. congesta  1.9180  1.8985  0.0030  0.4157  0.0007  0.7587  
Q. crenata  1.9030  1.8754  0.0085  2.8668  0.0020  0.7497  
Q. ilex subsp. ilex  1.4170  1.4302  0.0429  0.1534  0.0080  0.6133  
Q. suber  0.9535  0.9746  0.0615  0.2308  0.0151  0.7879  
Q. virgiliana  1.9440  1.9317  0.0015  6.6553  0.0003  0.6876  
Mean  Q. robur subsp. brutia  1.5500  1.6130  0.0493  1.6202  0.0129  138.47.00 
Q. cerris  1.7029  1.6773  0.0253  1.8675  0.0066  0.8327  
Q. congesta  1.8960  1.8815  0.0026  0.6569  0.0006  0.7587  
Q. crenata  1.8866  1.8675  0.0052  0.9650  0.0012  0.7497  
Q. ilex subsp. ilex  1.3625  1.3625  0.0053  0.4868  0.0010  0.6133  
Q. suber  0.9164  0.9323  0.0267  0.1453  0.0066  0.7879  
Q. virgiliana  1.9184  1.9144  0.0007  0.9555  0.0001  0.6876 
In the hypothetical case that the difference between the fractal values (means and medians) for two species is zero or has a quotient of one, the degree of relationship between the two species is 100%; DfA – DfB = 0; DfA / DfB = 1, species A and B are equal; thus the lower the fractal difference or the nearer the fractal quotient is to 1, the greater the similarity between the species.
The analysis of the FD values for each orientation and for each species shows that for Q. robur subsp. brutia, Q. cerris, Q. congesta and Q. virgiliana, the orientation influences the values of FD, as there are significant differences for these species (Table
KruskalWallis analysis for the values of FD in each orientation for each of the species. In bold: the significant values for which orientation influences the FD at 95% confidence.
KruskalWallis:  Q. robur subsp. brutia  Q. cerris  Q. congesta  Q. crenata  Q. ilex subsp. ilex  Q. suber  Q. virgiliana 

Mean North  1.5290  1.6676  1.8310  1.8669  1.3804  0.9001  1.9192 
Mean South  1.6220  1.6190  1.8749  1.8803  1.3442  0.9487  1.8780 
Mean East  1.7336  1.8110  1.9215  1.8476  1.3276  0.9059  1.9287 
Mean West  1.5676  1.6116  1.8985  1.8754  1.3895  0.9746  1.9317 
St. Deviation North  0.2702  0.1936  0.1174  0.1313  0.1723  0.2651  0.0406 
St. Deviation South  0.2992  0.1836  0.0763  0.1030  0.1220  0.2020  0.0777 
St. Deviation East  0.2069  0.1194  0.0288  0.1602  0.1074  0.2180  0.0563 
St. Deviation West  0.2829  0.2535  0.0546  0.0921  0.1564  0.2479  0.0392 
K (Observed value)  9.9875  20.5115  23.0332  1.6844  8.0795  3.0683  38.4400 
K (Critical value)  9.4877  7.8147  7.8147  7.8147  9.4877  7.8147  7.8147 
pvalue  0.0406  0.0001  < 0.0001  0.6404  0.0887  0.3812  < 0.0001 
These species correspond to deciduous or marcescent species, whereas the perennial species Q. ilex subsp. ilex, Q. suber and Q. crenata do not show significant differences in the values of FD for the different levels of orientation.
An analysis of the average FD values for each species indicates that there are significant differences between the different levels of species under study (Table
KruskalWallis test.
K (Observed value)  220.2702 
K (Critical value)  12.5916 
GDL  6 
pvalue (bilateral)  < 0.0001 
alpha  0.05 
Differences in FD by pairs between each species (in parentheses, pvalue). In bold: significant differences at 95% confidence.
Q. robur subsp. brutia  Q. cerris  Q. congesta  Q. crenata  Q. ilex subsp. ilex  Q. suber  Q. virgiliana  

Q. robur subsp. brutia  –  
Q. cerris  4.26 (0.6392)  –  
Q. congesta  71.21 (<0.0001)  66.95 (<0.0001)  –  
Q. crenata  68.55 (<0.0001)  64.29 (<0.0001)  2.65 (0.7439)  –  
Q. ilex subsp. ilex  58.63 (<0.0001)  62.9 (<0.0001)  129.85 (<0.0001)  127.19 (<0.0001)  –  
Q. suber  109.43 (<0.0001)  113.7 (<0.0001)  180.65 (<0.0001)  177.99 (<0.0001)  50.8 (<0.0001)  –  
Q. virgiliana  96.87 (<0.0001)  92.61 (<0.0001)  25.66 (0.001)  28.32 (0.0002)  155.51 (<0.0001)  206.31 (<0.0001)  – 
As can be seen in Table
The analysis of the medians of the seven groups (Figure
In the multiple comparison analysis (Figure
In the case of both mean and median values, it is confirmed that the value of the fractal dimension (FD) is less than 1.6 in the case of sclerophyllous Quercus and greater for marcescent and deciduous Quercus (Figure
The differences between average FD values for marcescent and deciduous Quercus species are very low (Table
Homogeneous character of the groups.
Species  Count  Sum of the ranges  Mean of the ranges  Homogeneous groups  

Q. suber  34  626.0000  18.4118  A  
Q. ilex subsp. ilex  59  4083.5000  69.2119  B  
Q. robur subsp. brutia  30  3835.5000  127.8500  C  
Q. cerris  30  3963.5000  132.1167  C  
Q. crenata  38  7463.5000  196.4079  D  
Q. congesta  37  7365.5000  199.0676  D  
Q. virgiliana  46  10337.5000  224.7283  E 
Based on the differences obtained from FDA–FDB = 0, the most closely related species are: Q. congestaQ. crenata 0.023; Q. cerrisQ. robur subsp. brutia 0.064; Q. virgilianaQ. congesta 0.033; Q. virgilianaQ. crenata 0.046; and Q. crenataQ. cerris 0.191. The most distant relationship is between Q. virgilianaQ. suber 0.982 and Q. congestaQ. suber 0.949 (Figure
There is a widespread consensus that complex objects with the same features can be included in the category of fractals. Selfsimilarity is one of the characteristics of fractal objects, meaning that when these images are broken down into smaller pieces, each one is identical to the whole. The fractional dimension is another of its features.
In the hypothetical case that the difference between the fractal values of two species is zero, or their quotient is one, the degree of relationship between the two species is 100%: Df_{A} – Df_{B} = 0; Df_{A} / Df_{B} = 1, species A and B are equal. Thus the smaller the fractal difference or the closer the fractal quotient is to 1, the greater the similarity between the species; if the value of this quotient is far from 1, as occurs between Df_{vi}/Df_{su} > 2, the species Q. virgiliana and Q. suber are very distant from each other. This occurs when the fractal values are the same and means that the same or similar characters have been measured
Finally, the orientation has no influence on the fractal dimension between either the same species or between the different species. This means that the shape of the distribution of the leaf vascular network is not affected by possible changes in orientation, thus discounting the effects of environmental variables such as amount of light, temperature, humidity etc., associated with orientation. This evidence is important in Quercus species, as in other cases, these environmental variables can influence seed germination and the capacity of some plant species to adapt to extreme environments (
We confirm that the application of fractal analysis identifies the phenotypical differences between species and can be used as a method to establish their degree of relationship; this is supported by molecular analysis by various authors. In this work we can affirm that sclerophyllous Quercus species have a fractal dimension of < 1.6 and marcescent and deciduous Quercus species have FD > 1.6; and that Q. crenata, a hybrid of Q. suber and Q. cerris, has a greater similarity to Q. cerris than to Q. suber. The low values of the mean and median FD revealed by the differences between the FD for marcescentdeciduous Quercus species suggest a high degree of similarity amongst the five marcescentdeciduous species. Based on their FD, marcescent Quercus species (semideciduous) are more closely related to deciduous than to sclerophyllous Quercus species, whereas the sclerophyllous Q. ilex subsp. ilex and Q. suber show substantial morphological differences with the marcescent and deciduous Quercus species, as evidenced by fractal analysis. These two species have followed different evolutionary paths from the others, as is to be expected, as the centre of origin of sclerophyllous Quercus species is Mediterranean, whereas deciduous Quercus species have a temperate origin and marcescent Quercus species come from the boundary between the Temperate and Mediterranean bioclimates (
We are very grateful to the anonymous referees and to Subject Editor Peter de Lange for their suggestions for improving the original article. This article has been translated by Ms Pru BrookeTurner (M.A. Cantab.), a native English speaker specialising in scientific texts.