rOL. 4, NO. 3 WATER RESOURCES RESEARCH JUNE 1968

Horton s Law of Stream Numbers1

A. E. SCHEIDEGGER

U. S. Geological Survey, Urbana 61801

Abstract. The possibilities for obtaining a rational explanation of Horton's law of streamnumbers are reviewed. The previous explanations of the stream-number law by (1) a growthmodel referring to generations of rivers and by (2) statistical graph-theory are compared.The graph-theoretical explanation seems superior to the other model of Horton's law, inasmuch as it produces not only the form of the law, but even, numerically, the naturally observed bifurcation ratio. It is also independent of the structural properties of the river net.(Keywords: Geomorphology; rivers)

INTRODUCTION

Some years ago, Horton [1945] stated im-)rtant laws concerning the average configura-p of streams in a drainage basin. All thesers are based upon the notion of 'order' of a

particular stream. Horton [1945] introduced aincept of 'order/ but his laws can be stated

equally well if a somewhat different concept ofr', due to Strahler [1957], is used. Accord-

lg to Strahler, first-order stream segments arelose that have no tributaries on a given map,jcond-order stream segments are those thative as tributaries only segments of first order,:. The term 'order' thus refers to individual/er segments, not to whole streams from head-iters to mouth. Thus, the confluence of arer segment of order n with another of order

Fproduces a channel of order n + 1; the con-[rence of a river segment of order n with one

lower order is simply ignored, i.e., it proves again a segment of order n. In any count-

procedure, a river segment is counted as aJgle segment between such junctions that pro-Ice a change in order.[Of the three laws of Horton, the first is con-

led with the topology of a river net, thelers with the metric. We discuss here only

topological law. It can be stated as fol-ra:

|The numbers n, of stream segments of orderpi a given drainage basin form, on the aver-

(ensemble average over many similar ba-

|xPublication authorized by the Director, U. S.leological Survey.

sins), a geometric sequence, so that

n,-+1 = arii

This law is called the law of stream numbers.The quantity Rb = 1/a = nt/i\i^r is oftencalled the 'bifurcation ratio.' It is interesting tonote that the numerical value of Rh seems tovary from 2 to 4, but is, for mature rivers, remarkably close to 3.5. Thus, Leopold et al.[1964, p. 13S] noted that, among many samplesof basins in the United States, the above figureof 3.5 is found.

Thus, considering the law of Horton statedabove, it becomes evident that a plot of streamnumbers on semilogarithmic paper against ordershould yield, on the average, a straight line.Making such a plot for a given drainage basinis called making a 'Horton analysis.'

Horton's law, as is implicit in the statementgiven above, has only statistical significance. Itwas found by Horton [1945] from purely empirical analyses of many stream basins. It is theaim of the present paper to review the rationalexplanations for it.

Horton's law as stated above is expressed interms of Horton orders. It is more convenient.for actual calculations, to use Strahler ordersinstead of Horton orders. In the Strahler schemeof ordering, orders are assigned to stream segments in the manner indicated above. Horton'slaw can also be expressed in the same fashionfor Strahler stream segments as for Hortonstreams, without making any significant error[Scheidegger. 19676]. Thus, in our paper, 'order'

655

656 A. E. SCHHIDEGGEK

always means Strahler order, unless anothertype of order is specifically mentioned.

HORTONIAN NETWORKS

The law of stream numbers discussed in thispaper is probably the most important of Horton's laws. As is evident from its formulation,it is concerned only with the topology of a rivernet; metric properties are completely left outof consideration.

Horton's law of stream numbers has givenrise to the concept of a structurally Hortoniannetwork. This is a network of, say, order Ar, inwhich the bifurcation ratio Rb = ni.1/nl forthe numbers nt of streams of order i is constant for all i < iV, and in which all complete,i.e. maximum, subbasins of order M < N havethis same property with the same bifurcationratio as that valid for the main basin.

Thus, a structurally Hortonion network ismade up of elements as follows: on the average,Rb first-order streams combine to form a second-order stream; on the average, Rb such second-order streams combine to form a third-order stream, etc. In such a network, one canspeak of 'cycles' or "generations' of streams. Thesubbasins referred to above are always taken toinclude all elements of a complete cycle.

It is clear that a structurally Hortonian network obeys Horton's law of stream numbers;however, such networks are not the only onesthat obey this law. It is evidently possible toconceive of a basin which obeys Horton's law ofstream numbers, but which is not structurallyHortonian. This was noted, for instance, bySmart [1967]. In such networks, it is no longerpossible to speak of 'cycles' or 'generations' ofrivers, so that the subbasins do not satisfyHorton's law of stream numbers at all, or notwith the same bifurcation ratio as the mainbasin. Such an example is given in Figure 1.

A test of a given network for structural Hor-tonianness has been given by the writer [Scheidegger, 19676].

CYCLIC MODELS

Although Horton only coimted stream numbers, Horton's law has generally been taken asimplying that river nets are structurally Hortonian in the sense defined above, so that onecan speak of "cycles' or 'generations' of rivers.

Horton himself [1945, p. 339] attempted a

Y

(a) (b)

Fig. 1. Two river nets with 0 first-ojjstreams, 3 second-order streams, and 1 third-orstream, so that the bifurcation ratio is corand equal to 3. However, (a) is structurallytonian, (b) is not.

hydrophysical explanation of his law of strnumbers essentially in terms of a growthcess, an idea which was formalized by Wolberg [1966], who tried to explain this lavthe expression of 'allometric growth' of asystem. Allometric growth means that thetive rate of growth of a part of the systemconstant fraction of the relative rate of gro,of the whole system. Woldenberg then intimatthat the law of stream numbers really shotuse as 'order' a quantity x which is notorder N itself but is equal to the bifurcati^ratio R„ to the power N — 1

x = (l/a)""1He referred to x as the 'absolute order' ofriver segment under consideration; it would |representative of the 'size' of the riverat the point under consideration. If this '(lute order' is used, Horton's law is expresseda power function

n(x) == axwhich is characteristic of allometric growl

The allometric growth process of ^berg can be further specified in various wjThus, for instance, one can envisage a bitand-death process that leads from generatto generation of rivers [Scheidegger, 1966].]is clear that any model in which the proc|leading from one generation or cycle of riverthe next is the same for every order will pi]duce a structurally Hortonian networktherewith Horton's law of stream numbers.

Horton's Law 657

lowever, no such growth processes can[sount for the possible existence in nature of

In-Hortonian networks. It is not clear howallometric growth process can be defined

en a given, river net is not structurallyDnian, so that one cannot speak of genera-

is or cycles of rivers. The whole allometric&wth-modei is dependent upon the idea thatly structurally Hortonian networks shouldst in nature.

<GRAPH ENSEMBLE STATISTICS

lorton's law of stream numbers can also beained by a statistical model that is not basedthe concept of cycles or generations of

rers. Thus, the writer has proposed [Scheideg-, 1967a] that a particular river network isrealization of a particular graph among all

sible graphs (arborescences) with the sameabers of pendant vertices (first-orderBams). The only reasonable assumption re-Wmg the probability distribution, for graphsrborescences) with a given number of pendant

Irtices is that every possible graph is equally|ely. This corresponds statistically to a 'micro-lonical' distribution: Every state of thetern that has a given value for the constant ofmotion involved (the number of pendant

tices) is equally probable in the ensemble ofpossible states. In the paper cited, the possi-graphs (arborescences) up to an order of

rere enumerated; for graphs with a higherpber of pendant vertices, a Monte Carlolod was proposed [Scheidegger, 1967a; Liao' Scheidegger, 1968] to sample the conditions.

j*he ensemble approach does not make thelption that river nets are structurallyonian; in fact, among the possible graphsare certainly graphs that are structurally

Hortonian. These are counted in andr̂aged.j?he result of using this procedure is startling,

only does one obtain. Horton's law ofeam-order numbers, implying a roughly con-it expectation value for the bifurcation ratio

jtere is a slight decrease with order), butexpectation value for the bifurcation ratio

[very close to that observed in the maturepers of the United States, viz., it is very close

3.5. This is a great achievement of theSsemble-statistical theory, inasmuch as theleories based on generation of rivers cannot

produce a numerical value for the bifurcationratio.

In context with the above, it may be notedthat Shreve [1966], in a discussion of Horton'slaw of stream numbers, also introduced thehypothesis of 'randomness' of the topologicalconfiguration of drainage basins. However, although Shreve calculated probabilities of occurrences of certain channel configurations, hedid not take the next logical step, involving theintroduction of a microcanonical 'ensemble' overwhich observable quantities have to be averaged,so that expectation values can be calculated inentirely general terms.

DISCUSSION

We have seen above that essentially twodifferent types of models have been advanced toexplain the existence of Horton's law of streamnumbers.

In the first tji^e, one assumes that all rivernets are structurally Hortonian. Horton's lawof stream numbers is then a direct consequenceof this assumption. In structurally Hortoniannets, one can speak of 'generations' of rivers,and an explanation is sought for the existence ofsuch 'generations.' Such an explanation is obtained by assuming that the conditions leadingfrom one 'generation' to the next are statisticallyinvariable.

In the second type of model, no hypothesesabout the intrinsic structure of river nets areformed, other than that such nets are simplytopologically bifurcating arborescences. If, then,nature operates completely at random, all possible arborescences must be equally probable.These arborescences may or may not be structurally Hortonian. This approach produces notonly Horton's law of stream numbers, but eventhe correct value for the actually observed bifurcation ratio.

In both models discussed above, one is dealing with statistical expectation values. However,it should be noted that the two possible typesof models do not produce the same statisticalpopulations: The cyclic type models produceonly a part of the theoretically possible bifurcating arborescences, namely only thosewhich are structurally Hortonian, whereas in theensemble-statistical approach the possible non-Hortonian networks arc counted in. It is evident,thus, that there is a fundamental difference

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A. E. SCUEIDEGGEK

between the two types of models, inasmuch asthe statistical populations are quite different inthem.

At this juncture, it is a matter of speculationwhich of the two possibilities is realized innature. Horton's law of stream numbers hasgenerally been assumed as equivalent to thestatement that river nets are structurally Horton nets, and Horton [1945, p. 339] has alreadyattempted to give a rational explanation of hislaw upon this basis. However, he already notedthe existence of 'adventitious streams' [Horton,1945, p. 341], which do not fit into this pattern.Thus, even early observations seem to favor thegraph theoretical approach. Theoretically, theredoes not seem to be any justification for theassumption that only those graph patternsthat are structurally Hortonian should benaturally possible. Surely, nature must be heldto be impartial and to produce all possible configurations equally often. The fact that thegraph-ensemble theory produces the correctnumerical value for the bifurcation ratio constitutes a further indication that this theory isprobably correct.

Therefore, there is certainly a conceptual, ifnot a factual (i.e. observational) reason toprefer the graph-theoretical explanation ofHorton's law of stream numbers over the others.

Acknowledgments. The writer is deeply indebted to Giorgio Ranalli of the University ofIllinois for drawing his attention to the fundamental differences between the 'graph-theoretical'and 'cyclic' approaches to Horton's law of streamnumbers.

REFERENCES

Horton, R. E., Erosional development of streand their drainage basins; hydrophysicalproach to quantitative morphology, Geol.Am. Bull., 56, 275-370,1945.

Leopold, L. B., M. G. Wolman, and J. P. MiPluvial Processes in Geomorphology, W.Freeman and Company, San Francisco, K

Liao, K. H., and A. E. Scheidegger, A compmodel for some branching-type phenomenihydrology, Intern. Assoc. Sci. Hydrol. E1988.

Scheidegger, A. E., Stochastic branching proceand the law of stream orders, Water ResouRes., 2(2), 199-203, 1966.

Scheidegger, A. E., Random graph patternsdrainage basins, Intern. Assoc. Sci. Hydrol., Ceral Assembly of Bern, Trans., HydrohiAspects of the Utilization of Water, pp. 417-1967a.

Scheidegger, A. E., Horton's law of stream mbers and a temperature analog in river iWater Resources Res., 4(1), 161-171, 19676.

Shreve, R. L., Statistical law of stream numb/. Geol, 74, 17-39,1966.

Smart, J. S., A comment on Horton's law of stnnumbers, Water Resources Res., 3(3), 773-1967.

Strahler, A. N., Quantitative analysis of watersgeomorphology, Trans. Am. Geophys. Un3S, 913-920, 1957.

Woldenberg, M. J., Horton's laws justified in teof allometric growth and steady state in osystems, Geol. Soc. Am. Bull, 77, 431-434, l!

(Manuscript received January 2, 1968;revised January 22, 1968.)