Robert W. Korn - Research Interests

1. Plant Development

My general research interest is in plant morphology and development with an emphasis on patterns. These patterns range from fundamental ones such as cell shape and stomata arrangement to the more superficial such as chimeras but are useful as probes into identifying the origin of form and pattern.

A. Chimeras

A chimera is a structure composed of genetically two different kinds parts. In humans a person with one brown eye and one blue eye is a chimera but in plants chimeras are more common and more variable. The type of plant chimera I’m interested in is a shoot periclinal chimera (two layers of different genotypes) that by a replacement division produces two daughter cells, one in each of two layers. Here a periclinal division in a mutant yellow tunica cell gives one mutant yellow daughter cell in the tunica and the other mutant yellow daughter cell in the green wild type corpus. This initial mutant cell in the corpus proliferates and its descendent cells take over the corpus. Thus there are three stages, (a) the original green shoot (corpus), (b) a yellow-green chimeric region where the yellow clone of corpus cells is beginning to take over the corpus and (c) finally a stable yellow shoot region when the yellow clone has taken over the corpus as shown in juniper on the right.

One feature of this phenomenon is that the majority (>80%) of chimeras become stable yellow and a minority (<20%) return to the green state. For the yellow clone to take over the corpus, and eventually the stem, requires that the initial replacement yellow cell was at the very top of the corpus and that there was a stability of location of cells and their progeny. These three conditions are met only when an apical cell resides at the top of the tunica that regulates growth rate and inhibits other cells from becoming apical cells (apical dominance in the plant physiology sense). This conclusion is radically different from the traditional view that chimeras come from three persistent apical initial cells at the summit in the tunica and three more in the corpus to constitute a multicellular meristem.

Korn, R. W. 2001 Analysis of shoot apical organization in six species of the Cupressaceae

based on chimeric behavior. Amer. J. Bot. 88: 1945-1952.

Another aspect of the chimera is the size and shape of the mutant yellow sector (arrow), together they tell something about the founder cell population that gives rise to a new leaf. Mapping the location of very small yellow sectors (1/24 of circumference of stem) suggests that the founder population is composed of about 170 cells, 70 of which are from the tunica and the rest from at least two layers of the corpus.

Korn R. W. 2002 Chimeric patterns in Juniperus chinensis “Torulosa Variegata”

(Cupressaceae) expressed during leaf and stem formation. Amer. J. Bot. 89: 758 - 765.

A third type of analysis of juniper chimeras concerns the length of the yellow-green chimeric region (region b above). The length ranges from 2 to at least 16 nodes with a peak at 3, expressing a discrete Pascal distribution or a continuous gamma distribution of stem length. This variability cannot come from growth of the apex where the yellow clone is passing down the corpus which then would also be highly variable in size so it is suggested that there is another,

basal meristem that produces leaves at a regular interval and at times causes the above apical meristem to make more basal meristem.

B. Tracheid analysis

Tracheids are xylem cells found in leaves and their distinct secondary walls of ring and spiral material have a different refractive index than water and hence are easily observed. Early in leaf development of the fern Thelypteris the midrib is composed of tracheids in tandem and the number of tracheids in the midrib is about eleven. Since the leaf develops at the tip the midrib lengthens eleven times each by the addition of a tracheid through cell enlistment. Tracheid analysis is then the scoring of tracheids for number, position, length, and shape.

Korn, R. W. 1998.Studies on the development of the leaf development of the fern Thelypteris

palustris Schott. Intern. J. Plant Sciences 159: 275-282.

More recently this approach has been applied to the leaf of the dicot, coleus and in particular the minor veins. It was found that first, the minor veins are composed of mostly three, sometimes two or four (figure below) tracheids in tandem. If there are three tracheids in a vein, or side of an areole, then the original areole was 3 x 3 or 9 cells. When a new minor vein, of three cells, form this leaves two daughter areoles of (9-3)/2 or of about 3 cells each. The three cells proliferate into a nine-celled areole that then forms another minor vein of three tracheids. Second, tracheid analysis of early leaves (19mm long) reveal isolated tracheids, a feature indicating a minor vein begins as an isolated tracheid that enlists one cell to either side to form a complete vein of three tracheids.

This small region of only 9 cells at which an areole begins is not that different from midrib extension in the fern leaf of about a dozen cells and for stomatal apparatus development of about a dozen cells, thus suggesting there is some unit size for patterning. This unit is termed a ontron (ontogeny = development,- tron = instrument). The ontron is not a formal structure like a cell or tissue but is a spatial requirement defined by cell number for development to proceed. One practical use is to see if there are enough cells available equal to the number required for some feature to form such as gland formation on a stem, root hair in roots or leaf initiation.

C. Computer simulation of plant patterns

Patterns in plants in most cases can be quantified such as for cell shape by the probability distribution for number of sides, for cell size by a lognormal distribution and for stomatal arrangement by the R value of Clarke and Evans. Once the pattern has been described, a computer graphic program is written that generates a drawing of the final pattern and data is then collected from the drawing. The test then is to see if the correct pattern is identified the data collected from the computer drawing is similar to data collected from real tissues. This method forces a more objective description of pattern because data are collected not as ends in themselves but to satisfactorily fit theoretical data.

Korn, R. and R, M. Spalding (1973) The geometry of plant epidermal cells. New

Phytol72: 1357- 1365.

Korn, R. (1993) Evidence in dicots for stomatal patterning by inhibition. Inter, J.

Plant Sci. 154: 367 - 377.

Korn, R. (2001) The geometry of proliferating dicot cells. Cell Proliferation 34: 43

- 54.

D. Other Problems

Other interesting cases of form in plants have been investigated. The banding in zebra grass with green and yellow strips like racoon’s tail is most interesting. The width of the yellow stripes follows a normal distribution whereas the width of the green stripes generates an exponential distribution. Also, all leaves of a shoot have the same peculiar pattern suggesting it originates in a common structure to the leaves, namely, the apical meristem.

When attempting to computer model leaf growth, it was found that there must be a marginal meristem to give the species-specific shape of a leaf. But there are other features along the margin, i.e., spines, lobes, and red coloration, suggesting some marginal structure exists where meristems of various types associate. This structure is termed the marginal band, and some evidence for it has been found. Generally, upper and lower leaf epidermal cells are isodiametric while marginal cells are slightly elongated along the minor leaf axis in the ice plant and much more elongated along the leaf’s major axis in Arabidopsis. Hence, a new structure of the dicot leaf has been found.

Korn, R. 2003. The marginal band, a new structural component of the developing angiosperm leaf. Bot. J. Linn. Soc. 143: 21 - 28.

2. Hierarchies

1. Features and Types

An interest in hierarchies came from my interest in plant development. It is general knowledge that an organism is composed of organ systems, organs, tissues, cell, organelles, etc. giving the erroneous impression that different levels collectively are like a stack of pancakes. Little in the literature on hierarchies goes beyond this point (Whyte, 1969) but somehow its seems that there must be another step to go from levels to organization. To me the solution to this problem was advanced by Howard Pattee (1969) when he proposed that there are vertical constraints that hold members at different levels together. Following up on this idea I have suggested that what makes an organization a hierarchy is not just vertical constraints but descendent vertical constraints where a superior holds a group of subordinates in place

The idea of a physical constraint as developed by Ernst Mach around 1830 is an entity that limits the degrees of freedom of movement of another entity. For examples, an anchor restrains a boat to move only in a rotational manner and a paper weight limits all movement of a stack of papers. This idea can be extended into the non-physical world of human societies as a constitution limits the action of its citizens, an instruction manual limits the behavior of the assembler and a gang leader dictates the behavior of other gang members. A descendent constraint is where one member places more constraint on another as the other does on the constrainer such as the Sun constraints the Earth more than visa versa. A horizontal constraints is where both members constrain one another equally such as binary stars or firemen in a bucket brigade.

This descendent constraint feature leads to the view that there are several types of hierarchies, physical, chemical and control. Physical hierarchies include astronomical, fluvial, tree branching, as well as gangs and they are characterized by the strength of the descendent constraints decreasing down the hierarchy. The second type of hierarchy is found in the chemical realm both in the inorganic and biological worlds and is identified by the strength of the constraints increasing down the hierarchy. Finally, there is the control type of hierarchy found in engineering, enzymatic and hormonal aspects of biology as well as in insect and human societies. The strength of the descendent constraint is absolute as it is defined somewhere as to who dominates over whom.

With a hierarchy defined as a group of units held together by descendent constraints, the hierarchical description of an organization usually does not reveal the business end of the whole. For example, the pope and the CEO of General Motors oversee bishops and board members, respectively, but the business of the Church is the saving souls and for GM the saving of soles. Similarly, a business partnership and vigilantes both can have several members at the top that share in making decisions, one might make cream puffs and the other creams toughs.

This view of hierarchies seems to remove the most interesting aspect of organizations, but hierarchical features indeed have a life of their own full of important and often curious relationships.

2. Span

The span of control is a business term referring to the number of subordinates under a superior. This number can be large such as 91 cardinals under the Pope and very small such as one when Sherman Adams was President Eisenhower’s chief of staff. University classes may be as large as several hundred students and a traditional middle class European family has a father ruling over a wife who raises a single child. In most cases, however, the span of control is about three to five according experts in the military and business. A squad has about six privates and one general has about three officers under him.

What seems to be the determining factor for the span is the nature of the constrainees. There can be many cardinals under a pope because they are well educated, smart men who need little direction in how to do their business. Students in a large class are treated all the same, they are experienced in the business of classroom attendance, and relatively little is expected of them. By contrast, small spans are found in the chief of staff where everything a president does is transferred down through him. Small spans exist where each subordinate is treated individually in either having unique duties, as in deployment of privates at the front or in a graduate class, where each requires considerable time of the superior. Contradictions do exist such as in kindergarten class which may be as many as 25 when they should be around five or a university basketball coach who has four assistants (s = 4) overseeing a squad of 12 players (s = 3) where big salaries and small spans are the perks of the business.

It is possible to determine the ideal span by a little computer program. Consider a span of size s, the number of members at the bottom, N, and the number of levels through which decisions are made, l, then the total number of decisions (d) through a hierarchy is s l. The ideal s is when sl is at a minimum. But there are three ways to arrive at the minimum value of d. One is under conditions of uncertainty where one doesn’t have the idea of what is best until all possibilities are explored, such as which candidate is best for some task. A second condition is making decisions involving certainty, that is, finding the one your looking for, such as the game of twenty questions where on an average one only has to search through only half the choices to find the right one whereas under conditions of uncertainty all possibilities have to be inspected. A third system has time or duration as critical in finding the best option. The time required to make a decision is related to where in the hierarchy the decision is made. A decision at the bottom of the hierarchy can be made fast (stop the assembly line because an error is occurring) and takes a long time at the top (decision as how many sedans, convertibles and SUVs to make). This

duration factor is simply taken as the square root of span size at level i, or ( ). N is the number of individuals at the bottom level.

Span Vertical steps Decisions

Uncertainty Certainty

(s) l= log N/log s) d=sc d=c(s-1)/2 Duration

2 6.64 13.28 6.64 37.3

2..718 4.61 12.52

3 4.19 12.57 6.95 37.8

4 3.32 13.28 7.47 38.9

5 2.86 14.31 8.00 34.8

6 2.57 15.42 8.56 43.8

7 2.36 16.56 9.10 50.4

8 2.21 17.71 9.67 51.3

9 2.09 18.86 10.21 49.8

10 2.00 20.00 10.80 41.7

The computer model gives the fewest steps under conditions of uncertainty when span size is 2.718 or the value e, or 2 under conditions of certainty and 5 for the shortest time or duration.The value of 2.718 for the ideal span size can also be worked out with a little calculus.

since d = l∙s

and s^l = N or l∙ln s = ln N or l = ln N/ ln s

and set ln N to 1

then d = s/ln s or f(s) = s/ln s

and the first derivative is f ’(s) = (ln s-1/s)/(ln s)² = (ln s-1)/(ln s)²

f ’(s) is at a minimum or f ’(s) = 0 when ln s-1= 0

such that ln s - 1 = 0 when s = 2.718

This quantitative view of a hierarchy as the fewest steps to make a decision can also be used to inspect how efficient a hierarchy is compared to other, non-hierarchical systems.

Intervals for total communication

Size Ring Wheel Hierarchy (3 levels)

2 1 2 2

3 3 3 3

4 3 4 3

5 5 5 4

6 5 6 5

7 7 7 5

8 8 8 5

9 9 9 5

10 9 10 6

n n=s when s is odd n log ( n)/log(3)

n=s-1 when s is even

Here we see the basic reason for the prevalence of hierarchies, they are the most efficient type of group known for transmission of information.

3. Constraint strength

As noted above there are three types of hierarchies, physical with constraint strengths increasing up the hierarchy, chemical systems with constraint strength decreasing up the hierarchy, and control where strength is relative and dictated. The actual strength of constraints in these first two types of hierarchies is of value to know because it determines separateness and stability of these levels.

Astronomical. Galaxies are composed of subgalaxies from first to fifth order yet the explanation for this clustering phenomenon remains unanswered. There is also the question as to whether galaxies are hierarchically or horizontally constrained . Most likely it is a case of hierarchical constraints because galaxies are spatially discrete.

Although only one planetary system is known, its organization has been easily understood since the time of Newton. The gravitational force between two bodies is given by the equation f = Gm₁m₂/d² where G is Newton’s constant or 6.67 × 10 ^-11/kg², m₁ is the mass of one body, m₂ is the mass of another body, and d is the distance between the two bodies. For example, the gravitational force between the Earth and the Sun is

6.67 × 10^-11 ×(1.05 × 10²⁴) × (1.97 10³⁰)/(1.5 × 10 ¹¹ )² = 3.3 × 10 ⁴⁴/2.2 x 10 ²²

= 1.1 × 10²² newtons

The Earth being about 10 ^-6 the size of the Sun it responds to this force by orbiting around a less moveable Sun. In similar fashion the force between our Moon and the Earth is

6.67 × 10^-11 ×(7.3 × 10²²) × (1.05 × 10²⁴)/(3.9 × 10 ⁸ )²

= 1.9 × 10²⁰ newtons

and that between the Moon and the Sun is

6.67 × 10^-11 ×(7.3 × 10²²) × (1.97 × 10³⁰)/(3.84 × 1019⁸)²

= 1.9 × 10¹⁹ newtons

While the Moon is somewhat influenced by the Sun it is much more constrained by the Earth.

The important question is what must be the critical difference in mass between two bodies in order to establish a hierarchical relation. Two bodies of equal mass form a binary system where their orbits overlap and a hierarchical system appears when one’s orbit totally encompasses the other’s orbit. Simple model systems can be constructed and a somewhat hierarchical relation appears when the masses have a ratio of about 8:1 and a clearly hierarchical relation appears with a ratio of 10:1. With the mass of the Sun at 1.97 × 10³⁰ kg and that for all the planets and moons at 2 × 10²⁷, or a ratio of about 1000:1, our solar system seems to be quite hierarchically stable with three levels of organization, the Sun, the planets and their moons.

Branching systems. Another physical hierarchy is branching exemplified by trees and river systems.. The relative strength of a constraint of a mother branch on a daughter branch is the surface area of a cross cut through the daughter branch at the point of attachment to the mother branch. I have measured the constraint strength of two species of trees, southern magnolia and the willow oak. Both species give the same result of a doubling (factor of 2) up the hierarchy, much less than that for planetary systems. Hierarchical constraints of river systems are more difficult to measure. The strength of this kind of constraint is more the diameter of a daughter river at the confluence with its mother. Measurements of four order of rivers of the Dragon Lake region in China from a internet aerial photo give an increase of about 1.9, close to that of 2 for tree branches. Taking stream diameter as proportional to river length also leads to an increase factor of 1.9.

The chemical realm. The electromagnetic force holding electrons and protons together in an atom is the foundation of the chemical bond. The covalent bond holds two atoms together through sharing of opposite charges and has a strength of about 100 kilocalories per mole (kcal/mole).Through electron screening not all the plus and negative charges of a covalent bond are neutralized and about 5% of the charges remain to make hydrogen bonds. In a molecule of water an oxygen atom is covalently bonded to two hydrogen atoms with four residual weak charges left over, two positive and two negative, Weak charges of different water molecules forge weak hydrogen bonds that hold water molecules together. The strength of the weak hydrogen bond is about 5 kcal/mole, about the same force as that associated with the velocity of water molecules at room temperature, such that the weak bonds are broken as they are formed. Consequently, water exists as ‘flickering clusters’ of about 16 molecules at room temperature.

Here there are three hierarchical levels, atom, molecule and macromolecule. But the hierarchy continues to extend upwards. Droplets of water fall from the sky or from a splash and they have a ‘skin’ of molecules with only two of the four charges neutralized and a central region of molecules where all four charged sites can interact with those of other molecules. The ‘skin’ is the next higher level as it constraints the movement of molecule within. But still some weak charges remain at the surface that can interact with other entities with weak charges to form an even larger and higher level structure. Another chemical hierarchy is found in biology with entities associated with different levels, for example, nitrogen (atom), amino acid (molecule), protein (macromolecule), nucleoprotein (macromolecular complex), chromosome (organelle), etc. which will be inspected more closely later. Constraints at all levels above amino acid sequences are realized by the hydrogen (H - - O) and hydro-sulfide (H- -S) weak bonds with the progressively higher levelsl held together by fewer such bonds per unit volume.

Control systems. Periodic constraints are from control structures resulting in alternating states of ‘on’ and ‘off’ as well as in the case of modulation where ‘on’ exists to different degrees. These control constraints are found in human and insect societies, engineer gadgets as well as enzymes and hormones in biology. They are unlike the previously discussed systems having constraints from gravitation and electromagnetic forces in that here the strength is absolute and where the corresponding members fit into a hierarchy is dictated by design. Various boss constraints in human societies are known to all members, thermostatic control over temperature is located by how it is built and a hormone interacts over another molecule by chemical specificity designed in the genes through evolution. There are also controls over controls (adaptive control), time control over thermostatic control and allosteric enzymatic control. Controls also parallel structural levels such as amino acids are held together by covalent amide bonds (C- - N) but which amino acids are sequenced is under enzymatic and RNA controls.

Structural constraints. As structures become more complex so do constraints, in fact, constraints are no longer just forces but structures themselves. Examples of this kind of constraint include nails, screws, glue, weld, pegs and wire, among others. In all cases these structural constraints interlock parts according to Ohm’ Laws which states that two atoms, and therefore two structures, cannot occupy the same space so are derived from electromagnetic force. It is difficult to state the relative strength of these constrains but they seem to become weaker up the hierarchy. A house has a cement foundation that holds wall 2" x 4" s in place by large bolts that hold studs together by 16 penny nails that hold door and window frames in place by 6 penny nails, and so forth. The relative strength up the hierarchy is like the chemical and not the planetary hierarchies in that it is weaker up the organization and like rivers it seems to differ by a factor of about 0.5.

D. Levels

A real hierarchy can be examined for level and span size, and being a botanist the white oak tree will be used. One can then depict a hierarchy of their interest after seeing how this description is done.

Level Example Span(approximate)

Biome Midwestern deciduous forest

Population White oaks in Jefferson county, Kentucky 10,000

Deme White oaks in my neighborhood              100

Individual White oak next to my shed              100

Organ system complex Branching 2

Organ system Shoot (stem with its leaves)             20

Organ Leaf          10

Tissue complex Dermal (upper, lower, edge)             3

Tissue Epidermis 2

Cell complex Stomatal apparatus 10,000

Cell Guard cell 2

Organelle Chloroplast            10

Macromoleclar complex Chlorophyll-protein                    5,000

Macromolecule Chlorophyll (tetrapyrrole)             5

Molecule Pyrrole 4

Notice that some spans are enormous such as 10,000 stomatal apparatus on an epidermis and some are very small such as two guard cells per stomatal apparatus. It would be desirable to eventually include constrain strengths in this chart as number of chemical bonds per μm³. Notice also that additional layers are added to the traditional ones, such as deme, organ system complex and cell complex, in order to have a continuity of constraints from top to bottom. The reader is left to add more layers above biome and below molecule.

4. Practical uses of hierarchical principles

Several apparently intractable problems in biology and philosophy can be resolved by applying hierarchical principles. First, there is the problem of whether a unicellular organism such as a bacterium, the protozoan Paramecium or the alga Chlamydomonas, is a cell or an organism? Some say it is a cell because it has cytoplasm and a nucleus while others claim it is an organism because it has polarity according to where flagella/cilia are located and by directional movement it has front and rear ends. The solution is that it is an organism with two levels, that of an individual organism characterized by directionality and that of the cell with a nucleo-cytoplasmic relationship (replication, transcription and translation). The either/or interpretation of unicellularity is replaced with multi-layered organizational one.

Another problem in biology is the evolution of complexity, how do tissues, organs and organ systems arise in evolution and development. The answer comes from observing the structure of moss plants. What happens is a structure at one level splits into two levels with the upper level constraining the lower level. For example a simple stomatal apparatus is composed of two guard cell that are attached to epidermal cells. This level of guard cell splits by guard cell precursors dividing into guard cells and novel subsidiary cells that specifically support the guard cells over the stomatal cavity. This is reminiscent of Parkinson’s example of proliferating levels in a human organization. One complains of having too much work to do so is rewarded by the company giving him some assistants, so the complainer is now just the boss of knowing what to do and his other tasks of doing jobs is given to assistants at a lower level.

A third problem that hierarchies can solve is that of the emergence principle in philosophy. Complex systems often exhibit a creative ability to generate new capacities. As examples of emergence of new faculties is consciousness in philosophy, new discoveries in technology and even the origin of life in biology. The trick to emergence is in rearrangement of constraints. Guttenberg’s printing press came from putting together two well known devices, first the signet ring becomes the moveable type and the grape press is modified to become the printing press. Another example is when parts of mechanical doll can be rearranged into mantel clock. Consciousness can come about by rearrangement of (a) episodic memory which is that memory associated with one’s experiences at one level and (b) knowing there is a future at a higher level and when reversing positions episodic memory is expanded into one’s own future. Consciousness is an evolutionary adaptive facility to see oneself in possible future states so one can pick the best outcome for survival. Episodic memory is rearranged to include anticipation and becomes consciousness. Finally, the origin of life can be explained only in somewhat unsatisfactory general terms. It is though that the first form of life might have been ribonucleic acid (RNA) that had both genotype (information) and phenotype (function). This duality of RNA could have come about by rearrangement of base sequences to give it both replicating and enzymatic properties by chance alone, similar to chromosomal aberration known to occur today.

5. Other facets of hierarchies.

Several other modifications of hierarchies are found such as when tasks are shared. Warren McCollough (inventor of computer memory) called this situation a heterarchy. A superior can be more than one individual as in a partnership or the U.S. government with the legislaturee, judicial and the executive branches.

Second, a hierarchy is different from a ranking system but the two are often confused. Positions of runners in a race, steps in an assembly line or flow chart and professorial ranks are not hierarchical, they are horizontal relationships because there is no significant change in constraint strength along the sites.

Thirdly, a member at one level need not constraint only members at the next lower level but those at any lower level. The Mississippi River is a seventh order stream but connects to1200 rivers of first to sixth order. Also, a full general has a chauffeur at a level some five orders lower as well as his underling lieutenant generals immediately beneath him.

Fourthly, a hierarchy as originally conceived by the Roman Catholic Church was a system of priests with the bottom level of lay people not part of the hierarchy. Here they are included because a hierarchy is a set of descendent constraints by the superiors on subordinates and the lay are involved in these constraints and so are a proper component of a hierarchy.

Lastly, besides hierarchical, descendent constraints there are horizontal constraints that add to the stability of the hierarchy. Bricks in a walkway, planets influencing each other, playground children holding hands to form a ring and managers discussing their problems are all examples of entities held together by constraints of the same strength.

Finally, I leave you with four hierarchical jig-saw puzzles for you to evaluate accordingly.

Korn, R. W. (1986). Hierarchical aspects of plant morphology. In (G. Rozenberg and A.

Salomaa, eds) The Book of L. Springer-Verlag, Berlin, pp. 207-216.

Korn, R.W. (1993) Apical cells as meristems. Acta Biotheoretica 41,175-185.

Korn, R. (1994) Hierarchical ordering in plant morphology. Acta Biotheoretica 42, 227-

244.

Korn, R. (1999) Evolutionary origin of plant structure: some hierarchical aspects. (In: M.

H. Kurmann and A. R. Hemsley, eds) The Evolution of Plant Architecture.

Whitstable Litho Printers, Kent, pp. 23 - 34.

Korn, R. (1999) Biological organization - a new look at an old problem BioScience 49,

51-57.

Korn, R. (2002) Biological hierarchies, their birth, death and evolution by natural

selection, Biology and Philosophy 17, 199-221.

Korn, R. (2005) The Emergence Principle in Biological Hierarchies. Biology and

Philosophy 20:137 - 151.

Pattee, H (l969) Physical condition for primitive functional hierarchies. In L. L. Whyte,

A. G.

Wilson & D, Wilson (eds), Hierarchical Structures ( New York: American Elservier Publishing Company).