Abstract

Beginning with Leonardo da Vinci's assertion that trees conserve total cross-sectional area across every branching point, I tested ten species of trees in the vicinity of Princeton, New Jersey, to see if they do indeed adhere to the rule of conservation as asserted by the Italian master and those who followed him. Based on my review of the literature, I expected to find that trees would either conserve area or not depending on the porosity of their wood to water. To my surprise, I found that all ten species conserve cross-sectional area in approximately the same way despite large differences in porosity. In particular, their twigs roughly doubled in cross-sectional area across each branching while their larger branches approximated area-preservation, as Leonardo had predicted. Rather than precisely preserving area, the trees actually tended to increase in area ever so slightly as I moved from trunk to twig tips. For this reason, I describe a conical model of tree form originated by Horn (1998, in press), which may estimate the volume of a tree more accurately than the traditional cylindrical model.

 

Introduction

Leonardo's Place in History

Figure 1. In his notebook, Leonardo da Vinci made this sketch depicting the branching pattern of trees. He depicted that the total thickness of branches along each of the arcs would e qual the thickness of the trunk. (Richter 1939, plate 27)

Many observers of nature -some scientists, some poets, some both- have attempted to explain the complex structure of trees. One of the most perceptive of these descriptions was made in the 15th Century by Leonardo da Vinci who observed that"all the branches of a tree at every stage of its height when put together are equal in thickness to the trunk" (Richter 1939, see Figure 1). In essence he was saying that the structure of trees is simple;that the cross-sectional area of a tree stays the same from its base to the tip of its smallest branch. It was such a beautifully articulated observation that it became, for lovers of trees, the basis for our understanding of branching structure until today.

The Pipe Model Theory and its Critics

Leonardo's theory has not yet been explicitly tested. It was simply assumed to be true, redefined by modern scientists, and then built upon. Shinozaki et al (1964a & b) reformulated this theory by conceptualizing a tree as a bundle of unit pipes, each of which supplies a particular amount of leaf area with water. They found a direct correlation between the total cross-sectional area at any horizontal level and the leaf mass above that level. From that time on, the analysis of branching structure has, for the most part, shifted away from structural allometry (scaled physical measurements) towards vascular allometry based on this "Pipe Model Theory" of Shinozaki et al.

The Pipe Model has significant implications for conservation biology. One of its potential uses is for estimating the biomass of forests, a common practice among foresters interested in lumber as well as scientists who want to know about the dynamics of carbon in the ecosystem.

Figure 2. An illustration of a tree that is compressed into a cylinder so that we can estimate its volume.

The Pipe Model is intuitively appealing once we understand the basics of water transport in trees. Trees must draw tremendous amounts of water from their roots up to their leaves in order to transport nutrients, dissipate heat, and maintain turgor pressure for support. The driving force of this water pump is the transpiration that occurs in the leaves, which draws up a continuous column of water through the xylem vessels (Zimmermann 1983). Since a plant's competitive advantage depends in large part upon its success in fixing carbon (i.e. the production of sugar and wood), and carbon fixation is limited by water accessibility, it is reasonable to say that a plant's growth is limited by its ability to get water to its leaves. For this reason, there should be a close correlation between a tree's conductive ability and its leaf biomass or leaf number.

However, upon closer examination, the Pipe Model has proven inadequate to describe the structure of real trees, though its essence still serves as the starting point for such studies. In particular, criticisms have focused on two assumptions made by the model. Shinozaki et al assumed that the xylem vessels of trees act like ideal pipes and that flow through them follows Poisseuille's law (in Tyree & Ewers 1991) which, in essence, says that the flow through a pipe is proportional to its radius, raised to the fourth power (i.e. flow ( radius4). In other words, if we double the radius of a pipe, we increase the flow rate by a factor of sixteen. While Poisseuille's law does help us to understand the gross differences in flow between trees with different vessel sizes, it fails to describe the details of flow accurately for four reasons. First, vessels are not perfectly cylindrical. Second, there is too much variation in the size of vessels of a single tree to make such a sweeping generalization about all vessels. Third, vessels do not maintain a constant diameter along their entire length; rather they taper and expand in strategic places (Zimmerman 1978, 1983; Tyree & Ewers 1991; Tyree & Alexander 1992). Finally, vessels are not actually continuous at all; rather they are made up of shorter units that are juxtaposed end to end (see Figure 3).

Shinozaki et al's second assumption was that trees conduct water through all existing xylem vessels regardless of the vessels' ages. Much research has been done to show that this is, in fact, not true. When a tree no longer has any need for its old vessels, or when they become too costly to maintain, it seals them with various resins and gums, forming what are called tyloses (pronounced ty-LOW-sees) (Zimmermann 1983; Cochard & Tyree 1990; Tyree & Ewers 1991). This phenomenon is familiar to anyone who has sawn through red cedar and seen and smelled its resinous heartwood. In some species, the vessels are sealed, not because the tree seals them physiologically, but because circumstances demand it. In these trees, air comes out of solution when the water in the xylem freezes in the winter, leaving deadly air bubbles called embolisms. Once such air bubbles form, the tree can no longer use the embolized vessels unless it can reabsorb the bubbles. If not, it must make a new set of pipes for the coming year (Tyree & Alexander 1993; Sperry et al 1994).

At the other extreme are trees with extremely small vessels such as sugar maple (Acer saccharum), tuliptree (Liriodendron tulipifera), and American beech (Fagus grandifolia) (see Figure 3). The water in the vessels of these trees also freezes, but their small diameter allows only minute bubbles to form. Most of these small bubbles are easily reabsorbed in the spring because of the very same positive pressure that allows us to tap for maple syrup (Zimmerman 1963). These trees can, therefore, recycle their vessels from year to year, meaning that their functioning sapwood is large relative to their heartwood. To this recycled sapwood, they add a fresh layer of xylem annually. Since their functioning sapwood is spread evenly throughout the diameter of the trunk, these trees are termed "diffuse-porous."

The final class of trees, which includes conifers such as white pine (Pinus strobus) and red cedar (Juniperus virginiana), is one in which continuous vessels are wholly absent. As depicted in Figure 3, the vessel elements of these "non-porous" trees are actually spindle-shaped cells that are arranged so that their porous surfaces are aligned with each other. These trees most likely conduct water much like diffuse-porous trees in that they use multiple rings of xylem tissue for transportation. The main reason for this is that only tiny embolisms can form in their vessel cells, and these can be easily reabsorbed in the spring.

Embolisms are not the only challenge facing trees. If we return to Poiseuille's law, we see that the ring-porous trees, because of their large vessels, are much more efficient than diffuse-porous trees. However, the increased efficiency incurs two major costs (Zimmermann 1983). It takes greater tension to pull water up the larger vessels. This increased tension raises the likelihood of cavitation (collapse and rupture of the vessel), which permits embolisms to form, thereby destroying the vessel irreversibly. In addition, the wider vessels are more susceptible to fungal infections such as the notorious Dutch elm disease. It is thought that these are the two most salient constraints on the diameter of vessels.

Modifying the Pipe Model

This difference in vessel conductivity between ring-porous and diffuse-porous trees means that the model proposed by Shinozaki et al (1964a) cannot accurately portray reality for all trees. In the wake of this realization, biologists have developed allometric models that take these different conductivities into account (Dvorak et al 1996; Gilmore et al 1996). Zimmermann (1978) first developed the concept of Leaf Specific Conductivity (LSC) to quantify the relationship between the volume rate of flow through a cross-section of xylem and the unit mass of leaves supplied by it. This differs from Shinozaki et al's model in that it takes into account the differential conductivity of xylem rather than assuming that all xylem conducts equally. Subsequent research has found that there is a large difference between the LSC of diffuse-porous and ring-porous trees (Zimmermann 1983; Chapman & Gower 1991). In particular, ring-porous trees, with their large vessels, have higher LSC values, meaning that they conduct more efficiently.

Even within individual trees, LSC differs for various parts of the tree (Zimmermann 1978). Specifically, the conductivity of the main trunk tends to stay constant while that of the branches decreases as we follow them from their junction with the main trunk out to their twig tips. This gradient facilitates an even distribution of water to all parts of the tree. Some describe this phenomenon through the analogy of an open basin in which all the twigs of a tree are dipped (Tyree & Ewers 1991; Yang & Tyree 1992). Branch junctions also tend to have a very low conductivity. These "junction constrictions" are thought to be protective measures to prevent excessive water loss from broken limbs (Zimmermann 1978).

The LSC correlation was modified further when it was found that not all sapwood is conductive, especially in ring-porous trees (White 1992; Chapman & Gower 1991). White discovered that the best correlation was to be found between the leaves and what he called "current sapwood area" (CSA), which amounts to the growth rings of the latest year and a half.

Putting Leonardo to the Test

These burgeoning discoveries about the hydraulic architecture of trees have led to the current study, which examines Leonardo's area-preserving hypothesis while considering differences in conductivity within individual trees and among various species of trees. Since all of the evidence cited so far indicates that the hydraulics of a tree impacts greatly on its architecture, it seems reasonable to say that Leonardo's area-preserving rule should also be contingent upon hydraulics. In particular, I expect to observe two pattern;one within individual trees and the other among species with differing vessel structures.

Within a given tree, I anticipate a slight downward taper in total area as I move from the main trunk to the twig tips. This trend is predicted by the original Pipe Model (Shinozaki et al 1964a). The vessels leading to branches that have been pruned or broken become heartwood prematurely, thus adding to the girth below a node without actually connecting to any leaves (see Figure 4). This means that the girth below every node should be slightly greater than the girth above the node, even in an area-preserving situation (Nygren et al 1993). This tapering effect could be compounded by the vessel constrictions mentioned earlier, for in order to mitigate the risk of cavitation in distal branches, each individual vessel tapers (Zimmermann & Potter 1982). I expect that this taper will manifest itself in the gross structure of the tree as well.

When I compare among various species, I expect to see differences that correspond to differences in vessel structure. Namely, I anticipate that the twigs of trees with thick piths like black walnut (Juglans nigra) and white pine (Pinus strobus) will overshoot the area-preserving rule because the pith comprises a major proportion of the cross-sectional area. The conductive xylem on the other hand comprises a small fraction of this cross-section. By extension, I expect trees with little or no pith to fall closer to the area-preserving rule. As I move to larger branches, I expect the split to be based predominantly on vessel size. Diffuse-porous trees--sugar maple, American beech, and tuliptree--and non-porous trees--white pine and eastern red cedar (Juniperus virginiana)--should preserve area across branchings because much of their wood is conductive sapwood. Conversely, I expect that ring-porous trees--namely oaks, American elm, and black walnut;will preserve circumference rather than area, since their conductive xylem is limited to the outer rings. According to this scheme, the only trees that ought to follow Leonardo's rule for all branch sizes are those that are both diffuse-porous and have no pith. A species with a large pith should grow twigs that appear above the curve and should grow large branches that settle into either the area-preserving pattern or the circumference-preserving pattern, depending upon whether it is diffuse-porous or ring-porous, respectively. The rest of this paper is full of surprises!

The breathtaking woods surrounding Princeton University in Princeton, New Jersey, served as my laboratory throughout this study. Many measurements were taken from trees in the woods of the Institute for Advanced Study and the Stony Ford Center for Ecological Studies. The more elusive American elm was observed along the walkways of the university itself, under the gaze of curious onlookers. Throughout the project, I discovered my primate roots as I climbed many a tree.

Fancy Equipment

For the most part, I gathered data using a simple tape measure and a caliper. I measured trees as high as I could reach them, whether by stretching or climbing. I found that recently fallen trees were a blessing in disguise, for I could take a myriad of measurements from the base of the trunk to the very tips of twigs in no time at all. But waiting for trees to fall is somewhat tedious. Fortunately, my labors were eventually eased when I was bestowed with a hand-crafted, home-made, top-of-the-line pentaprism caliper. With this optical caliper, I could measure trees quite accurately from a distance, without waiting for them to fall and without regressing to apish antics.)

My Subjects

I observed ten species of trees for this study. Since I wanted to compare species with different vessel structures, I chose several trees from each class (i.e. ring-porous, diffuse-porous, and non-porous). I also chose two species that have thick piths so that I could compare them to those that lack such a pith. Table 1 summarizes this information.

Table 1. A summary of the ten species that are included in this study along with there vessel structures and an indication of the presence or absence of a pith.

Measurements

I limited my measurements to even bifurcations where it was clear that one branch had split into two. It was important to avoid taking measurements at junctions where an old branch had broken off or had been pruned away (refer back to Figure 4 for explanation). It was also important to take my measurements at a reasonable distance from the junction itself because there was generally a considerable swelling above and below each junction. I found it safe to take these measurements at a distance from the node where the swelling had tapered down but before the swelling for the next junction began.
At each branch junction, I took three measurements: the diameter of the stem below the node and the diameter of each of the ramifications of that stem (see Figure 5). Using these data, I was able to analyze the branching structure according to the scheme laid out in the following section.

If we now conceptualize the radii a, b, and c as the sides of a triangle (see Figure 6), we find that the area-preserving rule recalls the Pythagorean Theorem with legs a and b and hypotenuse c. The circumference-preserving rule describes a distorted triangle where the angle between sides a and b is 180o (i.e. a straight line). Finally, the doubling rule produces an equilateral triangle, where all three sides are equal. To determine which rule a branch follows, we can look at the cosine of the angle between sides a and b (i.e. Cos of angle C). This angle can be described by the following equation:

According to the various rules, these are the values of Cos (C):
Area-preserving rule Cos (C) = Cos (90o) = 0
Circumference-preserving rule Cos (C) = Cos (180o) = -1
Doubling rule Cos (C) = Cos (60o) = 0.5
By graphing Cos (C) against the radius or diameter of the stem below the node, I will be able to determine which rule various sized branches tend to follow.

All ten species that I observed followed a nearly identical pattern for branching allometry throughout the tree. There was a tendency for twigs to follow the doubling rule, where Cos (C) = 0.5. This trend was not limited to the pithy white pine and black walnut, as I had predicted. Rather, it describes the trees with the thinnest twigs, American beech (see Figure 7), as well as the thickest, black walnut (see Figure 8).

Figure 7. Even though American beech has no pith, it demonstrates the same pattern as black walnut, where the twigs tend towards the doubling rule (Cos of angle C = 0.5), while thicker branches tend towards the area-preserving rule.

Figure 8. I expected black walnut to show a unique pattern because of its thick pith. While its twigs do tend towards the doubling rule (Cos of angle C = 0.5), black walnut is not unique in this respect (Horn, personal communication, also in press, 1998).

For branches greater than 5 cm in diameter, the scatter decreases and a constant trend emerges. For white oak, this trend is precisely area-preserving (see Figure 9). For all other species, the trend is slightly above the area-preserving line, implying that cross-sectional area is actually increasing gradually across every branching from the trunk to the twig tips (see Appendix A for graphs of all ten species and Appendix B for a complete listing of my data).

Figure 9. White oak shows the strictest adherence to the area-preserving rule of all the species studied (Horn, personal communication, also in press, 1998).

Putting it All Together

The Pipe Model Falls Short

A purely hydraulic explanation of tree allometry is not sufficient to explain the trends observed in this study. If, as I had hypothesized, vessel architecture played a significant role in dictating allometry, the outcome would have looked very different. In particular, the trees would have segregated into two groups based on different vessel structures and arrangements. Data from diffuse-porous trees would have fallen along the area-preserving line, while ring-porous trees would have dropped below this line toward the circumference-preserving line. In addition, the twigs of trees with thick piths would have overshot the area-preserving line and reached toward the line of the doubling rule.
If vessel architecture had played a larger role, I would also have found widespread scattering of the data within each species' data set. I expected this scatter because the data were collected from conspecifics in wide-ranging habitats. As I asserted earlier, tree growth is highly dependent on the environmental conditions. For example, a tree that grows in the midst of a dense forest would tend to grow in height faster than in width because it must quickly reach the light in the canopy if it is to survive. By extension, an open-grown tree will have the luxury of growing in more even proportions, since it is not constrained by light. There was no evidence of such a difference in my observations.
I expected an even stronger confounding effect to stem from the different abilities of these trees to cool themselves when subjected to direct sunlight. Trees with large vessels are severely limited in their cooling capabilities because rapid transpiration would likely cause very high negative pressures in the xylem, thereby increasing the risk of cavitation (Zimmermann 1983). If this is true, than oaks and elms grown in full sun might be less healthy than their forest-grown brethren. The lack of any such variation nudges me to look elsewhere for explanations.

Figure 10. This conical model is similar to the cylindrical model in Figure 2. However, this model may describe tree form more accurately or at least give us an upper estimate of the volume. Note that the extreme taper of the cone is an exaggeration.

For all the other species I examined, the cylindrical model will be an underestimate of the actual volume, since their total cross-sectional area actually increases from the base of the trunk to the twig tips rather than remaining constant. For these trees, the shape of the tree-space can be better described by a cone (I shall call it a pseudo-cone) whose point is buried under the ground and whose mouth flares upward toward the sky (see Figure 10) (Horn, in press 1998).

We can attempt to calculate the volume of this pseudo-cone using the insights gained from this study. The geometry is a little more complicated than it was for the simple cylinder, so I will relegate the details of that calculation to an appendix (see Appendix C). This conical model is most certainly an improved estimation of the volume of a tree, but it also has its limitations. One of these limitations lies in the fact that most trees have a relatively lengthy trunk that does not branch until it reaches the crown. The failure of the trunk to branch along much of its length means that its cross-sectional area is not increasing constantly as the conical model predicts. On the contrary, the trunk tapers from its base to the first branching (McMahon 1975). For this reason, the conical model overestimates the volume of the trunk itself, and by extension, overestimates the volume of the entire tree.

While both the cylindrical model and the conical model of tree form are estimates for the volume of trees, they each serve a valuable purpose. The cylindrical model provides us with a lower bound for our estimation of the volume. Using this value, we can approximate the lower limit of carbon storage for a stand of woods very easily. The conical model, on the other hand provides an upper limit for the same measurement. Even if these calculations seem more complicated than the simple cylinder, it is still a tremendous improvement over the alternative, which involves murdering the tree to weigh its dismembered parts. I anticipate that further study will enable us to extract the precise shape of this cone so that we can narrow the margin between the upper and lower bounds. The end result will be more accurate estimations of the volume of trees than those calculated by Nygren et al (1993).

In addition to the steady expansion in cross-sectional area that the lower parts of each tree exhibited, the twig tips consistently expanded at an extraordinary rate, many times approaching the line for the doubling rule. The fact that all species showed an identical trend in the expansion of total cross-sectional area at the twig tips indicates that the pith has no significant impact on allometry, contrary to my hypothesis. On the other hand, this expansion in twigs does corroborate White's conjecture (1993) that it may be necessary to have a minimum quantity of wood for a twig to be viable. This is plausible if we assume that the leaves' demand for water is proportional to their area. The leaves supported by a particular first-year twig are supplied by only a single growth ring. After this first year, however, the previous year's rings can be recruited to conduct as well‹even in ring-porous species (White 1993; Ellmore & Ewers 1986). Therefore, after the first year, it suffices to produce slightly less sapwood, while the first ring must be relatively large in order to sustain the foliage.

The most peculiar result of this study is the finding that there is no detectable difference among the allometries of trees with disparate vessel architecture. The lack of corroborating evidence goads me to look to other possible explanations for the pervasive patterns I found. Such an explanation may lie in a structural model of tree design (McMahon 1975; McMahon & Kronauer 1975). McMahon proposes a model of a tree in which the main consideration is the structural support of the individual branches and of the tree as a whole. This model shows that trees seem to increase their girth in response to mechanical stresses caused by bending under their own weight. In other words, increased width is a protection from buckling. But it is not clear from their model whether this necessarily implies that area should be preserved throughout the tree. Further inquiry into McMahon's "axe-head" model is needed.

A Summary

From the results of this study, it appears that the dimensions of the ten species I observed nearly obey Leonardo's rule and preserve area across every branching. The mystery that remains is the explanation for this phenomenon. The vascular explanations that have been presented are not adequate to explain my findings, so we must look to more universal ecological pressures to which the trees could be responding. A structural approach looks promising, however, the mechanics of wood are not yet understood clearly enough to realize this goal.

References

Chapman, J.W. & S.T. Gower (1991). Aboveground production and canopy dynamics in sugar maple and red oak trees in southwestern Wisconsin. Canadian Journal of Forest Research 21:1533-1543.

Cochard, H. & M.T. Tyree (1990). Xylem dysfunction in Quercus: vessel sizes, tyloses, cavitation and seasonal changes in embolism. Tree Physiology 6:393-407.

Dvorak, V., M. Oplustilova, & D. Janous (1996). Relation between leaf biomass and annual ring sapwood of Norway spruce according to needle age-class. Canadian Journal of Forest Research 26:1822-1827.

Ellmore, G.S. & F.W. Ewers (1986). Fluid flow in the outermost increment of a ring-porous tree, Ulmus americana. American Journal of Botany 73:1771-1774.

Gilmore, D.W., R.S. Seymour, & D.A. Maguire (1996). Foliage-sapwood area relationships for Abies balsamea in central Maine, U.S.A. Canadian Journal of Forest Research 26:2071-2079.

Horn, H.S. (in press 1998). Twigs, trees, and the dynamics of carbon in the landscape. In J. Brown & G. West, eds. Scaling in Biology. Santa Fe Institute and Oxford University Press.

Kemp, M., ed. and translator, & M. Walker, translator (1989). Leonardo on Painting. Yale University Press: New Haven, CT.

McMahon, T.A. (1975). The mechanical design of trees. Scientific American 233:92-101.

McMahon, T.A. & R.E. Kronauer (1975). Tree structure: designing the principle of mechanical design. Journal of Theoretical Biology 59:443-466.

Nygren, P., S. Rebottaro, & R. Chavarría (1993). Application of the pipe model theory to non-destructive estimation of leaf biomass and leaf area of pruned agroforestry trees. Agroforestry Systems 23:63-77.

Rennols, K. (1994). Pipe-model theory of stem-profile development. Forest Ecology and Management 69:41-55.

Richter, J.P., ed. (1939). In The Literary Works of Leonardo da Vinci, vol. 1. Oxford University Press: London.

Robichaud, E. & I.R. Methven (1992). The applicability of the pipe model theory for the prediction of foliage biomass in trees from natural, untreated black spruce stands. Canadian Journal of Forest Research 22:1118-1123.

Shinozaki, K., K. Yoda, K. Hozumi, & T. Kira (1964a). A quantitative analysis of plant form‹the pipe model theory. I. Basic analysis. Japanese Journal of Ecology 14(3):97-105.

Shinozaki, K., K. Yoda, K. Hozumi, & T. Kira (1964b). A quantitative analysis of plant form‹the pipe model theory. II. Further evidence of the theory and its application in forest ecology. Japanese Journal of Ecology 14(3):133-139.

Sperry, J.S., K.L. Nichols, J.E. Sullivan, & S.E. Eastlack (1994). Xylem embolism in ring-porous, diffuse-porous, and coniferous trees or northern Utah and interior Alaska. Ecology 75(6):1736-1752.

Tyree, M.T. & J.D. Alexander (1993). Hydraulic conductivity of branch junctions in three temperate tree species. Trees 7:156-159.

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White, D.A. (1992). Relationships between foliar number and the cross-sectional areas of sapwood and annual rings in red oak (Quercus rubra) crowns. Canadian Journal of Forest Research 23:1245-1251.

Yang, S. & M.T. Tyree (1993). Hydraulic resistance in Acer saccharum and its influence on leaf water potential and transpiration. Tree Physiology 12:231-242.

Zimmerman, M.H. (1963). How sap moves in trees. Scientific American (reprint) March, 1963.

Zimmerman, M.H. (1978). Hydraulic architecture of some diffuse-porous trees. Canadian Journal of Botany 56:2286-2295.

Zimmermann, M.H. (1983). Xylem Structure and the Ascent of Sap. Springer-Verlag: Berlin.

Zimmermann, M.H. & C.L. Brown (1974). Trees: Structure and Function. Springer-Verlag: New York.

Appendices

APPENDIX B
Measured diameters (cm) and calculated Cos (C)

Appendix B-1
Black and Red Oak (Quercus velutina and Q. rubra)Sheet1!R1C1:R61C5

Species	Dc (cm)	Da (cm)	Db (cm)	Cos C
BO	27.22	14.48	21.33	-0.12
BO	13.69	11.59	9.71	0.18
BO	11.01	9.26	6.27	0.03
BO	9.55	5.51	8.50	0.12
BO	6.91	5.54	5.35	0.20
BO	6.65	5.00	5.22	0.15
RO	59.30	41.80	41.40	-0.02
RO	54.60	40.50	37.80	0.03
RO	49.30	32.50	40.20	0.09
RO	47.70	34.50	31.30	-0.05
RO	38.80	30.50	17.70	-0.24
RO	36.10	31.40	21.70	0.11
RO	33.70	29.30	23.20	0.19
RO	33.00	25.10	20.10	-0.05
RO	31.80	25.00	23.60	0.14
RO	26.70	21.30	12.50	-0.19
RO	25.10	18.80	13.40	-0.19
RO	16.20	13.00	10.80	0.08
RO	14.30	11.80	8.90	0.07
RO	12.22	6.05	10.95	0.05
RO	10.89	8.91	7.07	0.09
RO	10.80	6.90	8.90	0.08
RO	8.66	5.22	7.70	0.14
RO	7.70	6.11	5.67	0.15
RO	7.60	6.00	5.70	0.16
RO	7.60	5.10	6.50	0.16
RO	7.45	4.55	6.02	0.03
RO	7.32	4.30	5.86	-0.02
RO	7.32	4.87	6.53	0.20
RO	7.30	4.30	5.60	-0.07
RO	7.13	4.52	5.98	0.10
RO	6.90	4.60	5.90	0.15
RO	6.53	5.06	5.22	0.19
RO	6.50	4.90	5.40	0.21
RO	6.40	4.60	4.80	0.07
RO	6.30	4.74	4.77	0.12
RO	5.80	4.60	4.10	0.11
RO	5.79	4.58	4.17	0.13
RO	5.19	3.57	4.20	0.11
RO	4.65	3.92	3.18	0.16
RO	4.50	2.90	3.80	0.12
RO	4.04	3.53	2.90	0.22
RO	4.00	2.90	3.30	0.17
RO	3.28	2.32	2.67	0.14
RO	3.15	2.26	2.51	0.13
RO	3.10	2.10	2.50	0.10
RO	3.02	2.32	2.39	0.18
RO	2.90	2.10	2.40	0.17
RO	2.40	1.60	1.90	0.07
RO	2.30	1.85	1.65	0.14
RO	2.20	1.91	1.88	0.33
RO	2.00	1.60	1.50	0.17
RO	1.40	1.10	1.00	0.11
RO	1.30	1.00	1.00	0.16
RO	1.20	1.00	0.95	0.24
RO	0.90	0.70	0.70	0.17
RO	0.90	0.65	0.60	-0.04
RO	0.70	0.50	0.40	-0.20
RO	0.55	0.40	0.50	0.27
RO	0.48	0.30	0.34	-0.12

We can find the volume of the pseudo-cone by finding the volume of the true cone that extends into the ground (see Figure 11) and then subtracting the imaginary part of the cone that is underground. The general formula for this difference of volumes is:

V = 1/3([ (D22)(H+h)/4 ­ (D12)(h)/4]
= 1/3([ (D12)(A1B-1)(A2)(H+h)/4 ­ (D12)(h)/4]

I shall break this formula down into its component parts to show its derivation (See Figure 11). I use V to mean volume of the tree; D2 is the diameter of the larger end of the cone, hence (D22/4 is the area of the large end of the cone; D1 is the diameter of the small end of the cone and is equivalent to DBH; H is the height of the tree; h is the height of the imaginary cone that extends into the ground; A1 is the average ratio describing the increase in cross-sectional area across each branching with the exception of the twigs; A2 is the same ratio as it applies to the increase in area of twigs; and B is the branching order, which Horn (in press 1998) describes as the number of branchings from the terminal twigs to the trunk.

Figure 11. A depiction of the conical model of tree form with its dimensions labeled.

Finding H and D1 is trivial, so I will not describe the process here. To calculate h, we can use similar right triangles to find that h = HD1/2(D2- D1). Since we cannot measure D2 directly, we must express it in terms of D1. To do this we must determine exactly how the diameter of the tree, or more precisely, how the cross-sectional area of the tree, increases from the base to the terminal twigs. This is why I introduce the terms A and B. We will define A1 as the average ratio of the total cross-sectional area above every node to the total cross-sectional area below every node except at the terminal twigs. In other words A1 = (a2+b2)/c2. A2 also equals (a2+b2)/c2, only it is specific to the terminal twigs (to understand why I must make this distinction between twigs and larger branches, refer back to the "Interpretations" subsection). So A, in general, describes the amount that the total cross-sectional area increases across each branching.

So we know that the cross-sectional area increases by a factor of A at every level of branching. And we know that branching order (B) quantifies the exact number of branchings for a particular species. So in general, AB gives us the ratio for the increase in cross-sectional area over B number of branchings. We can now focus this generalization to our specific case.
The area of the wide end of the cone is (D22/4. Since we cannot measure D2 directly, we must express this area in terms of D1 by using A and B. The area of the narrow end of the pseudo-cone, (D12/4, will expand into the area of the wide end by a factor of (A1B-1)(A2). The term A1B-1 takes care of the expansion from the trunk to the branchings just before the terminal twigs‹a value that seems to be about constant according to my findings. The term A2 takes care of the expansion at the twig tips, which tends to flare out more than the rest of the tree. Taking both expansions into account, tells us that the area of the wide end of the cone is (D22/4 = (D12(A1B-1)(A2)/4. This gives us all the information that we need to calculate the volume of this pseudo-cone, and by implication, the upper limit for the volume of a given tree.

I shall break this formula down into its component parts to show its derivation (See Figure 11). I use V to mean volume of the tree; D2 is the diameter of the larger end of the cone, hence (D22/4 is the area of the large end of the cone; D1 is the diameter of the small end of the cone and is equivalent to DBH; H is the height of the tree; h is the height of the imaginary cone that extends into the ground; A1 is the average ratio describing the increase in cross-sectional area across each branching with the exception of the twigs; A2 is the same ratio as it applies to the increase in area of twigs; and B is the branching order, which Horn (in press 1998) describes as the number of branchings from the terminal twigs to the trunk.