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{\Large Leaf Rank and Areole Size; the measurement, evolution, and ecology of dicot leaf venation}

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W. A. Green$^1$, S. Little, C. Price, S. Wing$^1$, L. J. Hickey, J. Weitz

$^1$National Museum of Natural History, Smithsonian Institution, Washington, DC, 20560

{\sc email} wagreen@bricol.net

Draft manuscript for AJB? Ann. Miss. Bot. Garden? IJPS?

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%AJB: http://www.amjbot.org/misc/ifora.shtml
%IJPS: http://www.journals.uchicago.edu/page/ijps/instruct.html
%AnnMissBotGar: http://www.mbgpress.info/index.php?task=instructions&article=10002

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{\it Abstract:}

Organization is an important characteristic of dicot leaf venation. It has been implicated in the origin of angiosperms, used as a feature to assist taxonomic identification of vegetative material, and related to leaf hydraulics, functional morphology, and growth form (habit). In this paper, we provide semi-quantitative data on the organization of 1730(???) genera of dicotyledons in 87(???) families as well as quantitative measurements of areole size for 100 specimens that were chosen to provide good systematic coverage across the angiosperms. We conclude that homogenously scored leaf rank is a useful proxy for areole size and variability, that areole size is essentially invariant across the leaf lamina and can be reliably measured on an area as small as 1 cm$^2$. It is also clear that leaf rank is significantly related to growth form and to drought tolerance and that at the level of Linnean family, variation in leaf organization (whether measured by rank or areole size) accounts for a large fraction of total variation across angiosperms. Leaf rank therefore remains a useful semi-quantitative proxy for leaf organization when areole size measurements cannot be made, as in the case of most fossilized material.

Keywords: leaf rank, venation, areoles, angiosperm evolution

Questions:\\
---how reviewy?\\
---include measurements made by hand?

\newpage

\section{Introduction}

The veins in vascular plant leaves distribute water and nutrients, transport photosynthate, and provide structural support and reinforcement of photosynthetic tissue, under the constraints of existing developmental pathways, and phylogenetic history. Leaves are one of the most commonly fossilized plant organs, and they often preserve vein networks in detail. Venation patterns are therefore potentially useful for taxonomic identification of fossils, for examination of evolutionary change through time, and for reconstruction of ancient environments.

In the past, however, it has proved difficult to quantify the venation even of modern leaves completely, or to characterize the relationships between specific aspects of a vein network and their associated functions or phylogenetic significance. Most earlier work, therefore, has either been qualitative \citep{vonettingshausen1861, hickey1971, hickey1973, hickey1974, hickey+wolfe1975, doyle+hickey1976, hickey+doyle1977, hickey1977} or focussed on single characters {CITE sack????}.

Even definition of precisely what a `vein' consists of can be controversial: all veins consists of some vascular tissue (tracheids), but must phloem be present, or can a single tracheid be considered a vein? Are fibers and parenchymatous bundle-sheath cells included? \citep{wylie1952}

The description of leaves by reference to their venation goes back at least to Linnaeus, who described, for instance, `nervis lateralibus' in the net-veined monocot,{\it Curcuma,} (`lateral veins'; what would be called actinodromous secondaries according to \cite{ellis+2009}) \citep[p. 2]{linnaeus1753} and `nervis transversalibus' \citep[p. 208]{linnaeus1753} in the dicot {\it Cerbera,}, literally `transverse veins', which seem to correspond to opposite percurrent tertiaries in \cite{ellis+2009}.

In particular, the presence of what what is typically called `multiple orders of leaf venation' is considered one of the most useful diagnostic features of the angiosperms. Only a few Instead of showing a random branching pattern or regular dichotomous branching, most angiosperm leaves show specific, organized patterns of larger and smaller veins running along defined courses.

One of the necessities in describing the venation of a leaf is some concept of the relative importances or sizes of veins: the course of the midvein is more important (both phylogenetically, hydraulically, and ecologically) than the course of any particular one of the finest freely-ending veinlets. The traditional way of doing this is to describe a leaf as having a certain number of orders of veination: a leaf with few orders of veination has veins all of roughly the same importance while a leaf that has many orders of veination has veins that fall into several more-or-less distinct classes. These classes can be defined by size of the vein or by its connectivity (e.g. relationship to the petiole).

%Morphological description of angiosperm leaves has always included some way of distinguishing the relative sizes or importances of veins. There seem to be two major reasons why this is needed: first of all, large veins matter more than small ones. That is, the course of the midrib or primary vein is more important in the description of a leaf than the course of any individual one of the finest freely ending veinlets. Note that this does not mean that the overall pattern formed by the larger veins is more important than the pattern formed by the smaller ones. The concept of vein order is generally used to describe the importance of each vein.

Classification of vein systems can be done mathematically (a typical reticulate or anastomosing vein network from a dicot leave can ususally be described as a {\it simple connected cyclic graph}, while dichotomizing veination like that in most ferns forms a mathematical {\it tree} {\it (simple acyclic connected graph)}.

The internodes or segments in such a graph can be named by their connectivity with the largest (midrib) considered primary, with higher orders increasing away from the petiole, or by considering the finest segments primary and counting connections up to the petiole. Vein segments can also be classified by gauge
(width, thickness, stoutness, and gauge are all synonyms for diameter in this context), by the developmental pathway that builds them, or by function.

Some of these methods of classification are more broadly applicable to other graphs found in nature:
for instance, in classifying streams and rivers, geographers call the smallest streams with no tributaries �primary� streams, those with only primariy tributaries �secondary streams�, those with only primary or secondary tributaries �tertiary streams�, etc. In the case of streams, assignment of order depends exclusively on connectivity or topology; regardless of a stream�s width, its order is one higher than the order of its highest order tributary.

%In the case of streams and rivers, relying on connectivity is sensible becasuse the branching pattern is exclusively dendritic at the scales being considered. Most dicot leaves, however, have a reticulate or anastomosing branching pattern, so the connectivity of a vein to the petiole is not well-defined. This defect is not entirely crippling as it would be possible to define connectivity along the shortest route to the petiole, but we have not pursued this possibility. Instead, we have looked exclusively at vein diameter in order to see whether a random sample of vein diameters shows distinct clusters or whether it forms an undiffernentiated continuum. In the former case, we may be able to define vein orders exclusively on the basis of vein thickness; in the latter, we must either reject vein orders as human constructs or redefine them to take less account of vein thickness.

In the human circulatory system, venation nomenclature is based on vessel diameter and histology. Veins and venules are distinguished loosely based on diameter, though there seems to be no rigid defining line between them, while a capillary is `a vessel whose wall consists of endothelium [a layer of flat cells lining the vessel] and its basement membrane; its diameter, when the c. is open, is about 8 mm.' A `postcapillary venule' is `the microvascularture immediately following the capillaries, ranging in size from 10 to 50 microns, and characterized by an investment of pericytes [connective cells generally found surrounding capillaries]' citep{stedman????}. Thus, the {\it definition} of vein orders is essentially functional (capillaries are differentially permeable; veins are impermeable), though they are generally recognized on the basis of {\it diameter.}

In dicot leaves, the most frequently used terminology is based on what is called vein order. This is a venerable but still somewhat vague concept, developed to distinguish `large' from `small' segments in a branching vein system.

Identification of the order of a vein in leaves is usually heuristic and imprecise, depending on both the connectivity of the vein to the petiole and on the diameter of the vein, though diameter generally trumps connectivity. That is, a very thin vein directly attached to the midrib is more likely to be classified by its size than by the fact that it is immediately connected to the midrib. Often, assignment of vein order is based on an implicit synthesis of these variables, especially in floras in which leaf description is only an incidental part of a more general entry. For instance, \citet{sargent1905} describes {\it Morus celtidifolia} as having `midribs and primary veins connected by conspicuous veinlets' \citep[p.305]{sargent1905} while  in the description of {\it Prunus americana} only `midribs and primary veins' \citep[p.512]{sargent1905} are mentioned. In other words, either two or three orders of veination are recognized, the first order being represented by the midrib, the second order by the primary veins (which would now be considered secondary), and the third order by the veinlets.

Up to to five orders of veination in certain leaves have been recognized for some time \citep{vonettingshausen1861}. \citet{ash+1999} recognizes `between four and seven orders of veination in most angiosperm leaves' according to the following prescription: All vein widths measured at their basal junction (i.e. thickest end). The thickest (largest diameter) vein is the main primary vein. The second-largest basal or approximately basal vein is considered primary if its diameter is greater than or equal to 3/4 the size of the main primary. If there are two or more primaries, then other basal or approximately basal veins are considered primary if their diameters are greater than or equal to 1/4 the size of the main primary. Secondaries are structural elements that do not meet the criteria for recognizing primaries or tertiaries. Tertiaries are the `widest veins that fill the field of the leaf' they `fill intercostal areas' and `cover the leaf lamina'. For higher ranks than tertiary, the same 3/4 rule at the basal junction applies. That is, where a vein branches sympodially (the smaller branch is greater than or equal to 3/4 the size of the larger branch) no higher order is recognized; where a vein branches pseudomonopodially (the smaller branch is less than 3/4 the size of the larger), a higher order is recognized. Higher orders are only recognized in anastomosing veins; all veins that branch strictly dichotomously are lumped into the highest order. 

Thus Sargent�s `midrib' is a `primary vein' in \citet{ellis+2009}, Sargent�s `primary veins' are now considered `secondaries', and Sargent�s `veinlets' are divided up into as many as five higher orders of veination. In this paper, we use the descriptive terminology in \citet{ellis+2009}, though our data suggest that a maximum of only three orders of venation are quantitatively distinguishable, even in the most densely and elaborately enervated dicot leaves.

In the case of leaf venation, no functional division of veins seems to have been attempted, though traditional developmental theory suggested that primary and secondary veination is laid down respectively by apical and marginal meristems while intercalary (plate meristematic) growth accounts for tertiary and higher orders. This distinction has been supported by recent developmental work \citep{poethig1984, piazza+2005}.

Another variable that could be considered is length, but this does not seem to have been applied in the case of leaf veination except by \citet{hill1980}, who distinguished secondary veins from `intersecondaries' using the product of their length and width.

\citep{wylie1938, wylie1939, wylie1943, wylie1946, wylie1949, wylie1951, wylie1952, wylie1954}

\citep{rothnebelsick+2001, rolland-lagan+2009}

\citep{melville1976}

%Although recent systematic work has established that early angiosperms were not taxonomically modern, the mystery of their rapid diversification and rise to ecological dominance remains. One correlate of their success was an increase in the complexity of organization: of flower architecture, leaf venation, pollination and dispersal ecology. I propose testing whether differentiation of the vein system in angiosperm leaves into multiple orders happened independently in multiple angiosperm clades, thus indicating convergent evolution, or whether it reflects the increasing dominance of a single clade that had the developmental machinery to be able to produce more differentiated vein systems. This research has the potential to explain the increase in organization in angiosperm leaves and relate it to the swift rise to ecological dominance of the angiosperms, which has been considered an `abominable mystery' since Darwin.

%In a 1879 letter to the botanist Joseph Hooker, Darwin referred to the `rapid development...of all the higher plants within recent geological times' as an `abominable mystery' (Darwin and Seward, eds 1903:20f). Despite over a century of exploration of `Darwin's abominable mystery', there remains little consensus about why the angiosperms were so wildly and swiftly successful.

The practice of assigning leaf ranks provides a subjective but relatively repeatable method of recognizing levels of organization in leaves: a leaf with few orders of veination and irregular vein courses is considered less organized than a leaf with highly regular veins falling into distinct classes. Organization, so defined, seems to correlate strongly with both phylogeny and ecology, but assigning leaf ranks relies heavily on the learned ability to recognize Gestalt categories. This perception that leaf ranking is a form of black magic has restricted its application.

In this paper, we describe a method for sampling vein diameters and illustrate how such a sample can be displayed using distribution plots of the log-transformed measurments to provide an alternative semi-quantitative description of leaf organization. From examining a few of such plots, we conclude that angiosperm leaves have predominantly two to five orders of veination and that the correlation between regularity of vein course and clustering of vein diameters is not always good. This suggests that a definition of leaf organization is needed that will distinguish between regularity of vein course and clustering of veins in size classes.

The second reason that it is important to distinguish between large and small veins is that the amount of distinction between them is itself an indication of the organization of the leaf. Biological organization is a difficult concept with which to grapple, but we have an intuitive sense that a leaf with many distinct size classes of veins is better organized than a leaf whose veins are all blurred together into a single continuum. This intuitive conviction that good sorting implies organization (and therefore biologically advanced organisms) appears frequently in evolutionary theory as, for example, in angiosperm floral evolution, where many undifferentiated floral parts are considered primative and small numbers of specialized parts advanced. This evolutionary progression from plastic generalists with many (possibly homeotic) similar parts to canalized specialists has been discussed by Gould�s (1987), and many other evolutionary theorists. The concept of leaf rank has been used to describe the organization of angiosperm leaves.

In this paper, we provide an alternate way to look at vein size and leaf organization based on the distribution of vein diameters. It is not intended as a replacment for vein order or leaf rank, but as a different point of view that may suggest insights into the ecology or development leaf veination.

Since the fossil record of the earliest angiosperms records changes in leaf architecture, I propose quantifying the concept of `organization' of leaf venation.

\section{Leaf Ranking}

The methodology I propose to employ is a development of the practice of assigning leaf ranks \citep{hickey1971, doyle+hickey1976, hickey1977, ash+1999}, which provides a subjective but relatively repeatable method of recognizing levels of organization in leaves: a leaf with few orders of venation and irregular vein courses is considered less organized than a leaf with highly regular veins falling into distinct size classes. Organization, so defined, seems to correlate strongly with phylogeny and ecology. For instance, Hickey (1974) pointed out that most taxonomic groups show higher organization in their more derived species and that the fossil record in general shows a gradient of increasing organization through the Cretaceous. Part of this trend may be due to local environmental and taphonomic bias (facies control) as fossil localities are predominantly from lowland floodplains and will show less variance than modern samples from a broader range of environments. Hickey's argument also relied heavily on the learned ability to recognize {\it Gestalt} categories, so without a reliable, repeatable, and precise method of quantification it is difficult to disentangle the influences of phylogeny and environment  on vein organization.

To do this, a precise definition of vein order is needed.

Regardless of how vein orders are defined, the number and distinctness of the different orders is also important in describing the veination pattern of a leaf and has received even less systematic consideration.

There seem to be few published alternatives to the qualitative description of leaves as `well' or `poorly' organized.

\subsection{Data}

The one we use here is called leaf ranking and was described in \citet{hickey1971, doyle+hickey1976, hickey1977} and subseqently included in \citet{ash+1999}. Leaf ranking involves assigning leaves to one of four arbitrary levels or ranks along a continuum of increasing organization. These four ranks (abbreviated 1r, 2r, 3r, and 4r) along with informal (hitherto unpublished) subdivisions (1r1, 1r2, 1r3, [2r0], 2r1, 2r2, 2r3, [3r0], 3r1, 3r2, 3r3, [4r0], 4r1, 4r2, 4r3; bracketed ranks are optional) give 12 to 15 ordered categories by which leaf organization can be subjectively but semi-quantitatively described.

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/Hickey1977Appendix.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

Hickey has used this descriptive method to explore the phylogenetic and ecological distribution of organization in angiosperm leaves: `Families tend to show one predominating level of organization. Families considered primative from other evidence cluster about the lower ranks with the intermediate and advanced families ranked higher, respectively. As might be expected, most families fall within the third rank. Similar, though smaller, differences in rank occur between the primitive and advanced genera of the better known families (Theaceae). The fossil record fo the angiosperms also shows a gradual increase in the rank of leaf assemblanges from the first to the third ranks during Early Cretaceous time....Finally, leaves of plants native to arid, arctic, or alpine environments frequently show a regression below the orgaizational level characteristic of the family.' \citep[][p.159]{hickey1977}.

This system, however, seems not to have been broadly applied because it relies, to an even greater extent than vein order, on the learned recognition of Gestalt categories. Therofore, it has only been possible to learn how to apply the system by calibrating one�s own perceptions of organization against a database of leaves with established ranks.

Under the current system described in the Leaf Manual, one assigns a vein order to each vein based primarily on the vein�s thickness and secondarily on its course. Then one ranks a leaf for degree of organization based on the distirbution of vein sizes and regularity of vein courses found in it. These assignments of vein order and leaf rank work provide a convenient conventional vocabulary for discussing veination pattern of leaves, but the system will always remain an arbitrary one. Such an arbitrary system for ordination works well for communication so long as each describer can calibrate his or her vein order and leaf rank assignments against the assignments of others. But because the system is optimized for communication, not representation, it reflects what we think we know and has little potential for overturning erronious assumptions.

In contrast, a set of more-or-less automatically obtained measurements, like a sample of vein diameters, is less effective for the purposes of communication than a descriptive vocabulary, but it may suggest patterns of vein organization or development that no qualitative vocabulary could record.

All leaves were scored by LJH (See Appendix 1) from herbarium specimens in the US National Herbarium.

\subsection{Analysis}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/FamilyBoxplots.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/RankHabitBoxplots.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/HSTHabit.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/RankDens.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/RankXericBoxplots.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\clearpage

\section{Quantification}

Relatively recent advances in computer modeling and fractal geometry have stimulated interest in synthesizing leaf-like networks relying only on simple physical models \citep[e.g.][]{bohn+2002, couder+2002, prusinkiewicz+lindenmayer1990, runions+2005, dimitrov+zucker2006}.

Physiologists interested in plant hydraulics have also measured `vein density' and pointed out its relationship to photosynthesis \citep{feild+2004, sack+holbrook2006, noblin+2008, feild+2009, boyce+2009}.

Visual perception of line thickness in known to be heavily subject to context. In order to avoid this sort of bias in assigning vein orders and leaf ranks, we measured the widths of pseudo-random samples of veins from several angiosperm leaves and examined their distributions to see how our heuristic ideas about vein order and leaf rank compare with numerical patterns in vein diameter distributions.

\subsection{Direct measurement}

I have made preliminary measurements of vein diameter distributions across leaf transects in six angiosperm species and compared their distributions with mixtures of standard normal distributions. These pilot measurements have empirically shown organization into 2--4 size clusters, depending on species. (See Figures 1 and 2.) This differs both from the 2--3 vein orders described in older floras like Sargent (1905) and from the 3--7 orders mentioned in the {\it Manual of Leaf Architecture} (Leaf Architecture Working Group, 1999). Combining these vein diameter distributions with angular distributions of vein course direction, I will acquire digital images of leaf vein networks from high-resolution digital photographs of 100--150 specimens of the National Cleared Leaf Collection, distributed across the five study clades. In addition I will acquire comparable images from a smaller sample of fossil leaves drawn from the Cretaceous collections at the NMNH. These will require hand tracing, but the number of fossil specimens of high enough quality for processing will prevent this from becoming an impossibly time-consuming task.

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/UlmusCinnamodendron.jpg} 
   \caption{Photographs of {\it Ulmus} (left) and {\it Cinnamodendron} (right) leaves}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/UCqqplot.pdf} 
   \caption{Normal quantile-quantile plots of the vein diameter distributions of the pictured genera. Note the step-like pattern in the {\it Ulmus} plot showing a greater clustering of vein diameters.}
   \label{fig:example}
\end{figure}

\clearpage

%The leaves examined were:
%									Specimen \#	reticle division
%Cinnamodendron cubense Urb			USNH2113817	?
%Drimys winteri (Winteraceae), cultivar		�			3.6 microns
%Maranta ruiziana Koern (Marantaceae)		NCLC1646	3.6 microns
%Ostrya virginiana (Mill.) K.Koch (Betulaceae)	NCLC6205	3.6 microns
%Phragmites communis Trin.(Graminae)		NCLC6345	3.6 microns
%Ulmus tomasi Sargent					USNH2230667	?

%In each leaf, vein diameters were sampled along one or more transects. In the case of the Maranta leaf, for instance, one transect (AA�) was perpendicular to and bisecting the midrib of the leaf, a second (BB�) was parallel and immediately adjacent to the midrib, and a third (CC�) was at an intermediate angle to the midrib (see Figure 2a). Obviously, in the case of the second transect the midrib was not sampled and all veins were measured at their bases. In the case of the Drimys leaf, the two transects (AA� and BB�) were both perpendicular to and bisecting the midribs and therefore should have similar distributions (see Figure 2b). The reason for sampling multiple transects on a single leaf was to show the robusticity of the patterns observed in spite of a defective sampling procedure.

%All vein diameters were measured with an eyepiece reticle in a compound microscope at 400 magnifications by rotating the eyepiece until the reticle was perpendicular to the vein being measured. The units in which the measurements are reported are reticle divisions and all measurements are rounded to the nearest unit. Obviously, this is not strictly a random sample. In the case of a pinnate leaf, the midrib or primary vein will be represented by the single largest datum. Secondary veins will have a broader distribution than they should if they taper away from the midrib and are measured at different distances from it. Rounding to the nearest unit truncates the measurements at a low level of precision, an effect that is particularly troublesome in the case of the finest veinlets. 

%Figure 2a: Leaf Transect and Vein Diameter Measurements: Maranta
%Figure 2b: Leaf Transect and Vein Diameter Measurements: Drimys

%Because of these defects in the sampling, it is difficult to interpret a simple histogram of the raw data (Figure 3a, 3b), so we also plotted the log-transformed data as histograms (Figure 3c, 3d), as scatterplots (Figure 3e, 3f), and as normal quantile-quantile plots (3g, 3h), in which each proportional fragment of the ordered sample is plotted against the equivalent proportional fragment of a theoretical normal distribution.

%Figure 3a: Histogram of raw Cinnamodendron diameter measurements
%Figure 3b: Histogram of raw Ulmus diameter measurements 
%Figure 3c: Histogram of logged Cinnamodendron diameter measurements 
%Figure 3d: Histogram of logged Ulmus diameter measurements 
%Figure 3e: Scatterplot of logged Cinnamodendron measurements against their rank order
%Figure 3f: Scatterplot of logged Ulmus measurements against their rank order
%Figure 3g: Normal quantile-quantile plot of logged Cinnamodendron measurements
%Figure 3h: Normal quantile-quantile plot of logged Ulmus measurements

%Interpretive annotations of the normal quantile-quantile plots vein diameter distribution plots, are given in Figure 4: the curves look like discrete step functions because of the excessive rounding error, but if we interpolate a continous curve we can see vein orders appearing in a set of rounded steps, which should not be confused with the steps due to rounding error. The overall sinuosity of the curve corresponds to the degree of clustering of vein sizes and can therefore serve as a rough guide to the organization of the leaf. The Cinnamodendron leaf was chosen as an example of a very low rank (disorganized) leaf; it would be considered about a 1r2 on Hickey�s scale and would have three orders of veination according to the Leaf Manual rubric. Our diameter distribution plots show only two orders of veination. The Ulmus leaf ranks at about 4r1 according to the Leaf Manual and shows a maximum of five orders of veination in the diameter distribution plots.

%Figure 4a: Normal quantile-quantile plot of Cinnamodendron data, transect A
%Figure 4b: Normal quantile-quantile plot of Drimys data, transect A
%Figure 4c: Normal quantile-quantile plot of Ostrya data, transect A
%Figure 4d: Normal quantile-quantile plot of Ulmus data, transect A
%Figure 4e: Normal quantile-quantile plot of Maranta data, transect A
%Figure 4f: Normal quantile-quantile plot of Maranta data, transect B
%Figure 4g: Normal quantile-quantile plot of Maranta data, transect C
%Figure 4h: Normal quantile-quantile plot of Phragmites data, transect A

%In addition to imperfect sampling, there are two other sources of error that may make interpretating these plots more problematical. First of all, primary and secondary veins frequently taper, so their diameter distributions may be more spread out than they would be if they were measured at the base. This source of error, however, is not a problem in the case of third and higher order veins, which do not taper systematically to any significant degree. Samples taken parallel to the midrib, like the Maranta BB� transect, clearly do not suffer from this source of error. Secondly, there is an irreducible degree of subjectivity about measurements made by eye: since veins are not entirely parallel sided and constant in diameter, there is a degree of approximation necessary in determining the diameter of a tapering vein. In the case of the larger veins, the proportional error is so small as to be negligable, but in the smaller veins this may lead to a recognition of imaginary vein orders. For instance, if most fourth order veins are between four and six units in diameter, one may unconsciously split too high a proportion of five-unit-wide veins into either four- or six-unit classes thereby making a single continuum look like two orders. This sort of bias could be avoided by automating the measurements or possibly by increasing the precision of the measuring technique, but so long as we treat the number of orders visible in diameter distribution plots like Figure 4 as the maximum numbers of orders that can be recognized, we are safe from this source of error.

%In all the leaves examined, there seems to be a normally distributed population of veinlets ranging upward from the minimum diameter (ignoring small outliers). In low rank leaves like Drimys or Cinnamodendron, this normal population (represented by a straight line on the normal quantile-quantile plots) may include all the veins except the midrib, which will be a single high outlier (Figures 4a, 4b). In high rank leaves like Ulmus, this normal population only includes a small proportion of the veins measured; most of the veins fall into more-or-less distinct size classes. Ostrya and Maranta, however, are relatively high rank leaves which seem only show three size classes of veination (Figures 4c, 4e).

%Maranta, in particular, is interesting because its veins have highly regular courses so it gives the impression of great organization. Extremely regular opposite percurrent tertiarys connecting the eucamtodromous secondaries give the sort of paxillate areolation that is usually restricted to very high rank (4r2 and higher) leaves. However, as can be seen from all three curves in Figure 5, it shows only three orders of veination.

%Figure 5: Comparative scatterplot of logged Maranta data for transects A, B, and C. Transect A is represented by �+�, B by ԥ�, and C by �*�. Note that despite differences in the curves, the same number of size classes (3) can be recognized in each curve. This shows the robusticity of determining vein orders from leaf transects: even in a highly anisotropic leaf like that of Maranta, number of vein orders is conserved across a ninety degree rotation.

%Another advantage of this form of measurement and representation is that it allows monocot leaves (like Phragmites) and even, potentially, ferns, to be compared directly with dicots. Though not intended to replace descriptive terminology, vein size distributions may prove easier than qualitative descriptions to interpret in the light of ecological, functional, or developmental theories. For this reason, we chose to examine Ostrya virginiana and Drimys winteri because both of these species have been the subject of some developmental work (Franck 1979, Gifford 1951). This will form the basis of further research.

%Finally, rigourous statistical comparison of our empirical vein diameter distributions with explicit probability models could be performed, but we question whether such a treatment would add anything to the insight that can be obtained from the diameter distribution plots of log-transformed data that we show in Figure 4.

\subsection{Image Analysis}

Using photographs of modern cleared leaves and image analysis software, we have succeeded in measuring a number of attributes of the vein network of more than 100 leaves, representing most major angiosperm groups and a few ferns. The measurements are repeatable and correspond to intuitive and descriptive notions of vein organization, like areole size and shape. We have observed significant relationships between venation attributes and ecological variables like growth form. Attributes of venation are also not randomly distributed phylogenetically. This supports the contention that leaf venation patterns encode ecological and historical signals, and gives promise that multivariate quantitative data will help disentangle them.

% schematic showing image locations and orientations
%\begin{figure}[ht] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=6in]{./figs/???.pdf} 
%   \caption{caption???}
%   \label{fig:example}
%\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/67.jpg} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/DataMedianQualityComp.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=6in]{./figs/TreeArea.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\clearpage

\subsection{Results}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/PairCompNCLC.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/MedIQR.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\begin{figure}[ht] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{./figs/ModeRank.pdf} 
   \caption{caption???}
   \label{fig:example}
\end{figure}

\clearpage

\section{Discussion}

Evidence from our measurements of vein diameters indicates that angiosperm veins mostly fall into one to three size classes, in comparison to the three to seven orders recognized in \citet{ash+1999} or the two to three orders recognized by, for instance, \citet{sargent1905}. Lower plants like ferns (pteridosperms) may only show only a single, diffuse order of veination.

%The last generation of botanists took for granted that the first angiosperms were woody, with showy, many-parted flowers like modern species of Magnolia. This story of floral evolution, the progression from many similar floral parts to fewer but more specialized components has come to seem grotesquely antiquated. So how did the highly specialized, organized structures of the angiosperms appear?

%Did they evolve on sunny, nutrient-rich, and heavily disturbed river banks (Hickey and Taylor ????), in alpine areas (???? ????), in the water (Dilcher ????), or in shaded disturbances on floodplains (Arens ????)?

%The concept of `organization' is a relatively vague idea that has nevertheless been important in past evolutionary thought, for supporting (or attacking) theories about the evolution of complexity and specialization, and notions of evolutionary directionality.

%Organization in leaf venation has been related to development, physiology, ecology, and phylogenetics. 

%This poor correspondance between different aspects of organization suggests that in our description of leaf organization we should try to deconvolute regularity of vein course from diameter clustering. The present study has presented a way of measuring and displaying diameter clustering; a similar procedure could be followed for producing distributions of vein course measurements and plotting them on rose-type diagrams.

%Such a quantification will allow application of McShea's (1994) terminology and his framework for distinguishing between `driven' and `passive' evolutionary trends (his `minimum' and `ancestor-descendent' tests will be applicable). These tests will allow determination of whether the observed increase in complexity, differentiation, and organization reflect real evolutionary innovations, in which many angiosperm clades figured out how to exploit their environments better (perhaps by occupying sites higher in the canopy with higher rates of photosynthesis), or whether they are merely the consequence of the increasing dominance of a clade that was successful for unrelated reasons.

%Specifically, I propose selecting a small number (20--30) of modern leaf specimens in each of 5 major sub-clades of angiosperms and determining whether quantitative increase in vein organization occurs within lineages (in parallel) or only in a certain group or groups. The preliminary selection of clades according to APG II (2003) is: 1. Nymphaeales, 2. Magnoliids (=Magnoliales + Laurales + Cannellales + Piperales), 3. Monocots, 4. Basal Eudicots (=Proteales), and 5. Higher Eudicots (=sister clade to Proteales). Sampling will be nested within clade to control for heterogenous variation.

%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=6.5in]{treefinal.pdf} 
%   \caption{Simplified relationships between potential study clades, according to APG II, 2003}
%   \label{fig:example}
%\end{figure}

%Each of these clades has identifiable fossil representatives in lower and upper Cretaceous rocks, spanning the time period during which previous work has suggested an increase in organization in angiosperm leaves (Hickey and Doyle, 1977). Deposits like the Potomac Group, Winthrop Formation, and Aspen Shale provide Aptian and Albian assemblages with large angiosperm components, while the Dakota and Meeteetse Formations provide Cenomanian and later Cretaceous examples; all of these formations are represented in the collections of the National Museum of Natural History, some very extensively. Fossils from the Potomac Group in particular, the largest collection of which is held in the NMNH, have been very influential in earlier discussions of angiosperm evolution, so the results of this research will be directly comparable to prior studies like those of Hickey and Doyle (1977).

%Parallel evolution of organization in more than one clade would provide strong evidence that lineages are responding to environmental conditions with an adaptive innovation (McShea's `driven trend'), while restriction of the trend to a single clade that increases in dominance would suggest a phylogenetic rather than an ecological explanation.

\section{Conclusions}

1. Leaf rank is inversely related to measures of areole size and areole variability.

2. Median areole size is essentially invariant across the leaf lamina and can be reliably measured on an area as small as 1 cm$^2$

3. Leaf organization (as measured by rank) is significantly related to growth form and to drought tolerance.

4. At the level of Linnean family, variation in leaf organization (measured by rank or areole size) accounts for a large fraction of total variation across angiosperms.

5. Leaf rank remains a useful semi-quantitative proxy for leaf organization when areole size measurements cannot be made, as in the case of most fossilized material.

\appendix
\section{Table of leaf ranks}
\begin{scriptsize}
\begin{tabular}{llcl}
\multicolumn{4}{c}{Table 1} \\ \hline\hline
{\bf Taxon}	&	{\bf Number}	&	{\bf Rank}	&	{\bf Habit}\\
\hline\hline
{\it Genus}			&????				&2r3	&xeric\\
\end{tabular}
\end{scriptsize}

\begin{thebibliography}{}

%APG (2003) An update of the Angiosperm Phylogeny Group classification for the orders and families of flowering plants: APG II {\it Botanical Journal of the Linnean Society} 141:399--436.

\bibitem[Ash, et al.(1999)]{ash+1999} Ash, A., et al. (1999) {\it Manual of Leaf Architecture.} Washington: Smithsonian Institution.
%Leaf Architecture Working Group (1999) {\it Manual of Leaf Architecture: morphological description and categorization of dicotyledonous and net-veined monocotyledonous angiosperms} Washington: Smithsonian Institution.

\bibitem[Bohn, et al.(2002)]{bohn+2002}Bohn, S., B. Andreotti, S. Douady, J. Munzinger, and Y. Couder (2002) Constitutive property of the local organization of leaf venation networks {\it Physical Review E} 65, 061914.

\bibitem[Boyce, et al.(2009)]{boyce+2009} C. Kevin Boyce, Tim J. Brodribb, Taylor S. Feild and Maciej A. Zwieniecki (2009) Angiosperm leaf vein evolution was physiologically and environmentally transformative {\it Proc. R. Soc. B}276:1771--1776

\bibitem[Couder, et al.(2002)]{couder+2002}Couder, Y., L. Pauchard, C. Allain, M. Adda-Bedia, and S. Douady (2002) The leaf venation as formed in a tensorial �eld {\it Eur. Phys. J. B} 28:135--138.

%Darwin F. and Seward A.C., eds., (1903) {\it More Letters of Charles Darwin: A Record of His Work in a Series of Hitherto Unpublished Papers, vol. II.} London: John Murray.

\bibitem[Dimitrov and Zucker(2006)]{dimitrov+zucker2006} Dimitrov, P. and S. W. Zucker (2006) A constant production hypothesis guides leaf venation patterning {\it PNAS} 103(24)9363--9368.

%Dengler, N. and J. Kang (2001) Vascular patterning and leaf shape Current Opinion in Plant Biology2001, 4:50--56 

%Nancy G. Dengler and Hirokazu Tsukayat (2001) Leaf Morphogenesis in Dicotyledons: Current Issues {\it International Journal of Plant Sciences}, Vol. 162, No. 3 (May, 2001), pp. 459--464

\bibitem[Doyle and Hickey(1976)]{doyle+hickey1976} Doyle, J. A. and L. J. Hickey (1976) Pollen and leaves from the mid-Cretaceous Potomac Group and their bearing on early angiosperm evolution. pp. 139--206 in Beck, C. B., ed. {\it Origin and Early Evolution of Angiosperms} New York: Columbia University Press.

\bibitem[Ellis, et al.(2009)]{ellis+2009}Ellis, B., D. C. Daly, L. J. Hickey, K. R. Johnson, J. D. Mitchell, P. Wilf, and S. L. Wing (2009) {\it Manual of Leaf Architecture} Ithaca, N.Y.: Cornell University Press.

\bibitem[Feild, et al.(2004)]{feild+2004} Feild, T. S., N. C. Arens, J. A. Doyle, T. E. Dawson, and M. J. Donoghue (2004) Dark and disturbed: a new image of early angiosperm ecology {\it Paleobiology} 30(1):82--107.

\bibitem[Feild, et al.(2009)]{feild+2009} Feild, T. S., D. S. Chatelet, and T. J. Brodribb (2009) Ancestral xerophobia: a hypothesis on the whole plant ecophysiology of early angiosperms {\it Geobiology} ???:???--???.

%Franck, D. H. (1979) Development of vein pattern in leaves of Ostrya virginiana (Betulaceae) Bontanical Gazette 140(1):77�83.

%Gifford, E. M. (1951) Early ontogenty of the foliage leaf of Drimys winteri var. chilensis American Journal of Botany 39:752�766)

%Gould, S. J. (1987) Time�s Arrow, Time�s Cycle Cambridge: Harvard University Press.

\bibitem[Hickey(1971)]{hickey1971} Hickey, L. J. (1971) [Abstract] Evolutionary significance of leaf architectural features in the woody dicots {\it American Journal of Botany} 58:469.

\bibitem[Hickey(1973)]{hickey1973} Hickey, L. J. (1973) Classification of the architecture of dicotyledonous leaves {\it American Journal of Botany} 60(1):17--33.

\bibitem[Hickey(1974)]{hickey1974} Hickey, L. J. (1974) A revised classification of the architecture of dicotyledonous leaves. pp. 25--39 in C.R. Metcalfe and L. Chalk, eds. {\it Anatomy of the Dicotyledons, Volume I, Second Edition} Oxford: Clarendon Press.

\bibitem[Hickey(1977)]{hickey1977} Hickey, L. J. (1977) Stratigraphy and Paleobotany of the Golden Valley Formation (Early Tertiary) of Western North Dakota {\it Geological Society of America Memoir No.150} Boulder, Colorado: Geological Society of America.

\bibitem[Hickey and Doyle(1977)]{hickey+doyle1977} Hickey, L. J. and J. A. Doyle (1977) Early Cretaceous fossil evidence for angiosperm evolution {\it Botanical Review} 43(1):3--104.

\bibitem[Hickey and Wolfe(1975)]{hickey+wolfe1975} Hickey, L. J. and J. A. Wolfe (1975) The bases of angiosperm phylogeny: vegetative morphology {\it Annals of the Missouri Botanical Garden} 62(3):538--589.

\bibitem[Hill(1980) ]{hill1980} Hill, R. S. (1980) A numerical taxonomic approach to the study of angiosperm leaves {\it Botanical Gazette} 141(2):213--229.

%Kaplan, D. R. (2001) Fundamental Concepts of Leaf Morphology and Morphogenesis: A Contribution to the Interpretation of Molecular Genetic Mutants International Journal of Plant Sciences, Vol. 162, No. 3 (May, 2001), pp. 465-474 

%McShea, D. W. (1994) Mechanisms of large-scale evolutionary trends {\it Evolution} 48(6):1747--1763.

\bibitem[Melville(1976)]{melville1976}Melville, R. (1976) The Terminology of Leaf Architecture {\it Taxon} 25(5/6):549--561. 

\bibitem[Noblin et al.(2008)]{noblin+2008}Noblin, X., L. Mahadevan, I. A. Coomaraswamy, D. A. Weitz, N. M. Holbrook, and M. A. Zwieniecki (2008) Optimal vein density in artificial and real leaves {\it PNAS} 105(27):9140--9144.

\bibitem[Piazza et al.(2005)]{piazza+2005} Piazza, P., S. Jasinski, and M. Tsiantis. (2005). Evolution of leaf developmental programmes. {\it New Phytologist} 167:693--710.

\bibitem[Poethig(1984)]{poethig1984} Poethig, S. (1984) Cellular Parameters of leaf morphogenesis in maize and tobacco pp. 235--259 in {\it Contemporary Problems in Plant Anatomy} White, R. A. and W. C. Dickison (eds.) Orlando, Florida: Academic Press.

%Price, C.A., and B.J. Enquist, Scaling Mass and Morphology in Leaves. (2007)  Ecology, 88(5): 1132--1141

%Charles A. Price, Brian J. Enquist, and Van M. Savage (2007) A general model for allometric covariation in botanical form and function PNAS 2007 vol. 104(32):13204--13209.

\bibitem[Prusinkiewicz and Lindenmayer(1990)]{prusinkiewicz+lindenmayer1990} Prusinkiewicz, P. and A. Lindenmayer (1990) {\it The Algorithmic Beauty of Plants} New York: Springer Verlag.

\bibitem[Rolland-Lagan, et al.(2009)]{rolland-lagan+2009} Rolland-Lagan, A.-G., M. Amin and M. Pakulska (2009) Quantifying leaf venation patterns: two-dimensional maps {\it The Plant Journal} 57:195--205.

\bibitem[Roth-Nebelsick, et al.(2001)]{rothnebelsick+2001}Roth-Nebelsick, A., D. Uhl, V. Mosbrugger, and H. Kerp (2001) Evolution and Function of Leaf Venation Architecture: A Review {\it Annals of Botany} 87:553--566.

\bibitem[Runions, et al.(2005)]{runions+2005} Runions, A., M. Fuhrer, B. Lane, P. Federl, A.?G. Rolland-Lagan, and P. Prusinkiewicz (2005) Modeling and visualization of leaf venation patterns {\it ACM Transactions on Graphics} 24(3)702--711.

\bibitem[Sack and Holbrook(2006)]{sack+holbrook2006}Sack, L. and N. M. Holbrook (2006) Leaf Hydraulics {\it Annu. Rev. Plant Biol.}57:361--81.

\bibitem[Sargent(1905)]{sargent1905} Sargent, C. S. (1905) {\it Manual of the Trees of North America} Boston: Houghton, Mifflin and Co.

%stedman???? Stedman's Medical Dictionary, 25th edition Baltimore: Williams and Wilkins.

%Miltos Tsiantis and Jane A Langdale (1998) The formation of leaves Current Opinion in Plant Biology 1998, 1:43--48.

\bibitem[von Ettingshausen(1861)]{vonettingshausen1861} von Ettingshausen, C. R. (1861) {\it Die Blatt-Skelete der Dikotyledonen} Wien: Druck und Verlag der Kais. K\"on. Hof- und Staatdruckerei.

\bibitem[Wylie, R. B.(1938)]{wylie1938} Wylie, R. B. (1938) Concerning the Conductive Capacity of the Minor Veins of Foliage Leaves {\it American Journal of Botany} 25(8):567--572.

\bibitem[Wylie, R. B.(1939)]{wylie1939} Wylie, R. B. (1939) Relations Between Tissue Organization and Vein Distribution in Dicotyledon Leaves {\it American Journal of Botany} 26(4):219--225.

\bibitem[Wylie, R. B.(1943)]{wylie1943} Wylie, R. B. (1943) The Role of the Epidermis in Foliar Organization and its Relations to the Minor Venation {\it American Journal of Botany} 30(4):273--280.

\bibitem[Wylie, R. B.(1946)]{wylie1946} Wylie, R. B. (1946) Relations between Tissue Organization and Vascularization in Leaves of Certain Tropical and Subtropical Dicotyledons {\it American Journal of Botany} 33(9):721--726.

\bibitem[Wylie, R. B.(1949)]{wylie1949} Wylie, R. B. (1949) Variations in Leaf Structure Among Adiantum pedatum Plants Growing in a Rock Cavern {\it American Journal of Botany} 36(3):282--287.

\bibitem[Wylie, R. B.(1951)]{wylie1951} Wylie, R. B. (1951) Principles of Foliar Organization Shown by Sun-Shade Leaves from Ten Species of Deciduous Dicotyledonous Trees {\it American Journal of Botany} 38(5):355--361.

\bibitem[Wylie, R. B.(1952)]{wylie1952} Wylie, R. B. (1952) The Bundle Sheath Extension in Leaves of Dicotyledons {\it American Journal of Botany} 39(9):645--651.

\bibitem[Wylie, R. B.(1954)]{wylie1954} Wylie, R. B. (1954) Leaf Organization of Some Woody Dicotyledons from New Zealand {\it American Journal of Botany} 41(3):186--191.

\end{thebibliography}
\end{document}