Network Analysis in Geography

332 Pages·1969·43.471 MB·English

by  Peter Haggett, Richard J. Chorley

NetworAkn alysiinsG eography
PETER HAGGETT
ProfeosfsU orrb aann dR egioGneaolg raip1ith hye
UniverosfBi rtiys tol
J.
RICHARD CHORLEY
LectuirnGe ero graipnth hyUe n iverosfCi atmyb ridge;
Fellofo Swi dnSeuys sCeoxl leCgaem,b ridge
EDWARD ARNOLD
�
-
SERIE
Preface
Ifb ookasr et raceatbhlreo ugmhi sttya xonomtirce etsod istainntt ellectual
forebeatrhse,nt his book cmliagihamtn cestirnyR oberHto rtonr'es ­
markablpea peorn t hee rosiodneavle lopmeonfst t reamTsh.i s1 94p5a per
bya nA mericacni veinlg ineleari tdh eb asinso to nlyo fm ucha nalytical
\Voriknfl uviagle omorpholbougtym oreg eneruanld erpinnifnogras p ro­
cesasp proacthoq uantitamtoirvpeh ologIyt.fis n dingwse ret aughttoo ne
of ubsy VaughanL ewis Caatm bridgaen d tot heo thebry Arthur
StrahlaetrC olumbifao;r b otho fu si tr epresenttheesd t arotft wot rails
ofw ork,t heo nem ovinga way frogme omorpholoignyt ol ocational
analysainsdt heo thetro wardtsh es tudoyf d rainabgaes inasst hef unda­
mentagle omorphuinci t.
Itw ast hec rossionftg h espea thast C ambridgient he easrilxyt tiheast
ledu st oc hecks omeo ft heo n-goiansgs umptionst heod ni chotoomfy
physicaanld h umang eograpahnyd t od rawt ogethtehres tranodfst his
bookI.n i tw e havet rietdoe xplowraey si nw hicht he analoyfas tiosp o­
logicadlilsyt icnlcatso sfs patisatlr uctures-lnientewaorr ks-mitghhrto w
lighotn commong eographpirco blemosfm orphometroyr,i gignr,o wth,
balancaen dd esigInt.b uildosn a ne arliveorl umbey o neo ft hea uthors
(Locatiaonnaalliy nsh iusm agne ograapnhdi yn) places incorporates relevant
sectiobnust,i tw identsh ea reao fs patiaanla lystiosi ncludpeh ysical
netwosr.kW e are tcooon sciooufst hed angerosf e asya nalogy and
strainmeedt aphorc ltaoi tmh atf,o re xamplset,r easmy steamnsd t rans­
port systemasr eg eographic'atlhley satmeod' o; s ow ouldf orcues t o
ignoraes pecotfsn etworskt ructuarnede volutitohna ta rei ntrinsically
importatnotp hysical haunmda ng eographreerssp ectivRealtyh.ew re
arguteh at thaerreee q uivalsepnatt isatlr uctucroemsm ont ob othf ields
andt hawte canu sec ommonm athematimcoadle lfso rb othk indHse.n ce
,,·lea yn o claimtso c ompletenoersr si gouwre: h aved rawn omno dels
fromo uro wna reaosf e xperiencet haemn adt hematilceavle ilse lemen­
taryW.e hopeh owevetrh ats omeo ft hep roblemrsa isewdi lalt tract
mathematicifaonrts h,e isro lutiwoinlsl surely adnee mlaengda nacned
powerb eyonodu rc ommandF.o rg eographweerh so pet hasto mef amiliar
materiwaillb le c asitn a newl ighatn dt hawte masyt arsto med iscussions
ona lternawtaiyvseo fv iewinogu rfi eld.
Thel engtohft hea cknowledgmesnetcst ioofnt his bsouogkg esstosm e­
thinogft hed ebtw eo wet oo therAs n.u mbero fw orkers, noCtharbilsyt ian
'\TerneMri,c haeWlo ldenberagn dO ve Pederseanl,l oweuds a ccestso
unpublishpeadp erNso.r thwestUenrinv erspirtoyv idae ds upertbr ans-
v
•
Vl PREFACE
portatiloinb raarnyd s timulatcionlgl eagfuoerws r itipnagr tof t heb ook.
Otherp artosft heb ookw eret riewdi thv aryinsgu cceosnsc olleagauneds
studenattCs a mbridgaen dB ristTohle.a ssistsatnataff t b othd epartments
helpeidn t hep reparatoifot nh eb ooka ndw e arep articularly tgor ateful
MargareRte ynoldMsa,r yN orcliffSei,m onG oddenT,o nyP hilpoatntd
RoberBti gnaflolr h elpa tv arioucsr ucisatla geOsu.r familibeys n,o w
hardenedt hteol ittoefrg alleayrso untdh eh ousea,s sisatneddr etarded
thep rojeicntc haracterfiassthiico Wne. d edicatthei bso okt oo urt wo
eldecshti ldriennt hef aint hopet htehyma aty sometimeensc ouratghee
otherisn tfhoer mearc tivity.
PETER HAGGETT
RICHARD CHORLEY
ChewM agna,S omerseStu.m mer, 1969
•
Chapteorn e TopologSitcr uctures
I.N ETWORKSA S GRAPHS.
IIB.R ANCHINGN ETWORKS.
i. Conceofp thsi erarocrhdie2cr. B. ranching
(aL)a wo f path n(ubRm)eb geirosvn.aa rli aitnbi iofnusr craattiioons .
ratios.
3.D atlai mitaftoirto onpso lcao ngailys4i.sS .t ochaasptpirco atcohn eest wotrokpl oogy.
I II C .I RCU IT NETW RK S.
0 r.E lementgarrayp h-thmeeoarseutri2.ec Bs i.n ary
(aO)r igicnoanln ectmiavtirti(ycb eP)so .w erceodn ­
connectmiavtirtiyc es.
nectivity( cWm)ea itgrhictcoeendsn .e cmtaitvriitcye s.
3.S hortesmta-tpraitche s.
(aC)o mparoifsg orna p(hbsC).o mparoifsn oond es and links.
IV.B ARRIERN ETWORKS.
r. Structruergaurlli ati2e. Esq.u ilibmroidueml s.
(aU)n bounndeetdts h:'e m id-contpirnoebnl(tebamB)l.o ' u ndneedtt sh:e
'islparnodb'l em.
Geographerresg ulaernlcyo untperro bleimnsv olvifnlgo wisn t heirre ­
searpcrho grammeCso.n sidtehrep hysicgaelo grapshteurd yisntgr eaming
ofw atewri thian d rainabgaes ionr t heh umang eograpshteurd yiinngt er­
regionsatlr eaomfsm igranBtost.h e xamplreesp resdeinstt ifnucntc tional
systebmust hatvheef undamental pirnocp oemrmtoynt hatth eiyn voltvhee
flowso fs omec ommoditthyr ouag hc hanneolra n etworokfc hanneTlhse.
probleimnso rganizfilnogwi sn teoffi ciecnhta nnepla tterns ainndt etrh­e
pretatiooftn hsed istinccthiavnen neelt wortkhsa hta vee merged-ofatse n
majofre atuornet sh ee arths'usr face-ftohremm a jotrh emoef t hivso lume.
Thea pproafcohl lowheedr ies t ob egiinn P arOtn ew itthh en etworks
ast heye xtlsta ndt oa nalytshee isrp atisatlr uctiunrt ee rmosf t opologic
2)
�Chap1.) a ndg eon1et(rCihca p. componenHtasv.i negs tablitshhee d
mains tructucrhaalr acteroifsc thiacnsn el networks tshhei fftioencd.u s is
ParTtw o tot heierv aluatiTohner. e lationne towfo rskt ructtuorf el ow
3)
demands( Chap. andt hep robleomf o ptimallloyc atinnegt works
1Chap4.) a ret hem ainc oncerinnt hisse ctiPoanr.tT hree analtyhsee s
growtahn dt ransformaotfn ieotnw orokvse tri me( Chap5.) T.h roughout,
thee mphasiso ni tsh eg enerparlo pertainedsp erformaonfcn ee tworks
andt hes patial prtohbelpyeo msseb ,u tt hesaer ei llustrwaitterhde ference
toa numbero fc ommonly-occunrertiwnogr sky steimnsp hysical and
humang eography.
I.N ETWORKS AS GRAPHS
Thet opolosgtircu ctoufra en etworikn volvtehser eductoifot nh ec hannel
pattetroni tmso stb asiacn de lementfaolr mW.h ethewre conceiav nee t­
,,�oirnkt ermosf a standadridc tiondaerfyi nitoif'o anm eshed faobfr ic
3
:-- .
A •I •• ,_ II •' \\ ' . \\ ,_ ...- ' ...... .. .• "' B
-.. •• .,..._ -- ...... ..--.. ··- -- •I y'I '.,• )" /II ,,_,_.' __ _J__-- .. -
·-
" ' >- --
•
.
•
� ' I
•' I I
I t I ,.
'r --� I- > ' --.....,'. II' ,I I'
I I I I
' •
� I.II ,...,. _J I_.".. ..,.III. _ _ I II \.""' _ .., ""' \\_ ..,\'\ I. ,'I"I .,. I ·•- --f'-,' "'l.I _ ,.. ,''_
.-.-:. -..-rr� ---:._-:.- :.-_ -._:-- :.:---·-=;.-:-- ..- .- ..� -� .. .......-._._ _-_ _.,,. ..�_. _ .l ......_I. -.1 r-I' -""'.. ',' ,, r JIItIII-4... - , -\ _.-.,·..' -""', ,.\ I'- '. ', _ .< --�.._.-I....I .. ,.. ' _·.,I.)-' ..,.,-. ,'.,.'.I', .. I ........ . ..-_....,.._'..,..... .!,.-..',• _' ' • 1,.. '.. ,, •.. ..., .,. .,.,_ -.,._ .....,.r _.,)• -,_. ,, -',I 'J..1 II., .. -,.,.'I,'\..! 'I .,.' . , .'..,.,, ' \.•" _., ,If _ ..,. (_- I f_...,- ,. ...r.l.I... I\ -,.-�__ f, r" ,,-.,_ ..J_'.III,.,' M ..,.I., '" ,,\ ,., ..' \,._, '.. �..tII ).' .., . ,-I '- · ,- -'..-.. ·__.•.:�, I · &.t,,. ( . ..JI·..I( .I I ' , -• \ ,I..,.II' _ ..� (\ , --'\\. J - ,�-_.. ..- ..-..,I'.. ,'1... ! ...i. ·
'A llAICY�\(.,. . .
0 .... MOULlRI( .D
.,. r J
1
A
·-
t::
FigI..I .S amplnee tworsky stem(sA.)C hannepla tteronfts h eM ississniepaprCi l arkesdMailses,U. .,S .Ab.r;a nc..h.
ing networkRso.a d(c Bo)m municatisoynsst efmo rs amea reaa sA ; circuit ne(tpwloarnkas(r C)).A irlipnaet tern
forF loridUa.,S .A.c;i rct1it n(entowno-rpklsa n(aDr)P) r.o perstuyb divissioountsh,e rn UO.hSi.oA b.a;r rineert ...
worksS.o urcMee:a da ndB rown1,9 62p.. 1 91T;h rower1,9 66.
I. I NETWORKS AS GRAPHS 5
intersecltiinneagsn di nterstoirci ensg 'e ographteresr'm ass ' as eto f
geograplhoicca tiionntse rconneicnat esdy stebmy a numbero fr outes'
)
(KanskIy9 ,6 3p,. r , we area utomaticcaolnlcye rnvveidt ha vercyo m­
plexb undloef· c haracteristhiocws :f artv hiezl .o catiaornesf romo ne
anothewrh,e thetrh er outjeosi nitnhge ma res traigohrct u rvewdh,a t
commodittyh en etworcka rriwehse,t hetrh efl owi sc ontinuoorui sn ter­
mittent,s oao nnd( FigI..1 )C.l earallytl h esaes pecatrse higrhellye vant
toi ndividnueatlw orbkust,t heayl smoa kec omparisboentsw eenne tworks
verdyi fficuletx cepatta triviaanlds elf-evildeevneItln .o rdetro g eta t
theb asiscp atisatlr uctouftr hee netwmourckho ft his informmuasttbi eo n
a
b
Es
Fig1.. 2R.e ductioofna map ofa transpnoerttw or(kA )t oa graph( B).
b
a
d
c
Fig1.. 3A.l ternattiovpeo lofgoircm fso rt heg rapmha ppedi nF ig1.. 2.
initidailslcya rdaeldt,h ouogfhc ourisetm ustb er eintrodluacteiedr nt he
analysis.
Int hifisr scth aptneert worakrser educetdo tlheev oeflg rapGhrsa.p hs
area rraoyfsp ointwsh,i calrie.c onnectoernd o t connetcoto endea nother
byl ineTsh.e ries n oc oncern twhiet hl enogrot rhi entaotfit ohnel inneosr
whethetrh eayr es traiogrhc tu rve(da ltholuignhem sa y bgei vepno sitive
orn egatisvieg nis.,e d.i rected orga rsaspihgsnn;eud m ericvaall ueis.,e .
valugerda phssot) h,a atn alyosfig sr aphs mraeyv ecaolm mont opological
structubruersi eidn a pparentulnyl iknee tworkFsi.g .1 .2 showst he
reductioofna simplreo adn etwor(kw iteha chn oder epresenatnin g
origiinn,t ersectteiromni,n oarls ,e ttlemoennt th er outet)oa systeomf
node(s vertVi1c-eVs1l )i,n k(se dgEe1s- E9a)n d( regiRo1n-sR 4S)i.n cwee
may disregdairsdt anacnedd irectiito npi oss sibtlore e -dratwh eg raph
6 NETWORKS AS GRAPHS I. I
(Fig1.. 2 -Bi)n sae rioefsa lternaftoirvmews h icsht iplrle sertvheeb asic
patteorfin n tercotninoencbse tweenno deasn dl ink(ss eFei g1.. 3)Because
.
a numbeorf g raph represeonfta a gtiivoennnes t worcka nb ef ormeidt,i s
moreu seftuols torteh ei nformataiboonu tth en etworiknm atrifxo rm.
TableI. I showas serioefsb inarmya tricfeosrn odesl,i nkasn dr egions
inw hichI represean dtisr eccotn nectiboent weetnh ee lemenatnsd o
connectivity incidence
otherwiTshee.s e matricmeasy bes upplemenbtye d
matricienws h icclo1n nectiboentsw eenno deasn dl inknso,d easn d regions
andl inla<nsdr egiomnasy ber epresenitnse idm ilfaorr m( Garrisaonnd
Marble, Ip7 p-.20 ).
Thes tudoyfn etworiknst hitso pologsiecnaslbe e gawni tEhu lerl'7 s3 6
papeorn t hes evebnr idgoefst heP russicaint oyf K onigsbearngd c on-
Table1. 1.C onnectivMiattyr icefso rG raphm appedi nF ig1.. 2.
Vert(Vi )c es Edg(eEs)
I 2 6 I 2 6 8
3 45 7 3 4 5 7 9
I 0 0 I 0 0 0 0 I 0 I I I 0 0 0 0 0
2 0 0 I 0 0 0 I 2 I 0 I I I 0 0 0 0
I I 0 I I 0 0 I I 0 I 0 I I 0 0
3 3
0 0 I 0 I I 0 I I I 0 0 I 0 I 0
4 4
0 0 I I 0 0 I 0 I 0 0 0 0 0 I I
5 5
6 0 0 0 I 0 0 I 6 0 0 I I 0 0 I I 0
0 I 0 0 I I 0 0 0 I 0 0 I 0 0 I
7 7
8 0 0 0 I I I 0 0 I
Regi(oRn)s
0 0 0 0 I 0 I I OJ
9
I 2
3 4
I 0 I I I
2 I 0 I I 1 = Connected
I I 0 I o Notc onnected
3 =
I I I 0
4
tinuewdi thC ayley('1 s8 79m)a p-colourpirnogb lebmu; tt hef ircsotm ­
prehenstirveea tmeonfnt e twortko polowgays n otp ublishuendt i1l9 36
Theodreieren dliucnhdue nne ndlGircahpehne n.
inK onig's The brancohf
topolodgeya liwnigt he lementasrtyr uctuwrhei,c hc amet ob e called
grapthh eory,
hasd evelopreadp idilnyt het hredee cadessi ncKeo nig's
seminwaolr ka ndi sn ows ummarizienad r angoef t ext(se .Bge.r ge1,9 62;
Busackaenrd S aaty1,9 65F;l amen1t9,6 3H;a raryN,o rmana ndC art­
wrigh1t9,6 5O;r e,1 963).
Oneo ft hec omplicatiinoa npsp lying grapht ott hheea onrayl ysis of
networskt ructure viesr yct ohnef usaendd o verlapptienrgm inology.
Lineasr ec ommonlrye fertroea ds' link'se'd,g e'ss'i,d e'sa'r,c' ss'e,g ments',
'branch'erso'u,t oers' 'o ne-cewlhlisl'pe;o intasr ed escribe'dn oadse s',
'verti'cjeusn'c,t i'oinnst'e,r sec'ttieornmsi'n,oa rl' sz'e ro-cIenltlesr's.t ices
withitnh en etworokr i nt hes urroundairnegaa ret erme'dr egioonrs '
'faceTse'r.m sa reo fterne stritcota e pda rticualpaprl ifieedl d( e.gi.nt, h e
I.I NETWORKS AS GRAPHS 7
medicalli teraltiunrkes naondde bse come 'neuarnodn' ss'y napsefso'r) ,
grapthh eorhya sf ounrde levanicnae varieotfyp roblemBse.s idtehse
obvioaupsp licattioto hnes desoifeg lne ctrical acnidtr ocfl uoiwtsts h,r ough
transportsaytsitoen(m Asv ondo-Bod1i9n6o2F,;o rda ndF ulkers1o9n6,2 ),
stanidt s uisnte hsed esigonf arte(fAalcetxsa ndI9e 6r4,o) r i nt hea nalysis
ofl inguissttircu ctu(rBeuss ackaenrd S aaty1,9 65)I.n anthropology,
sociolaongdys ,o cipasly choltohgeyr iesa t raditoifio tnus s ei nu nravelling
'cliqsuter uctuarnedsd 'e scribpiantgt eronfss o ciianlt eracwthiiocngh o es
(
backt oK urtL ewin'Psr inciopflt eosp olopgsyicchaoll o19gy3 6)L.i nks
betweesno cial intearnadmc otdieolnos fn euraalc tivhiatvyeb eenf orged
by Rapaporwth,i lues eosf d irectaendd v aluegdr aphhsa vee xtended
Table Topologcilcaasls ificoafnt eitowno rks
x.2.
Networks
Pla11Naert s N 011-PlaNneatrs
I
Paths Trees Circuits Cells
I
LineaBra rriers
LineaFrl owS ystems
\
SourceH:a ggett,C li1no rlaenyd H agget1t9,6 7p,. 1 0.
frome conomiocusti ntfioe ldsm aonfa geme(net. ign.t hed evelopmoefn t
'criticaaln-aplaytshai nsd'P ERT techniq(uBeast terIs9 b6y4,) ).
Treatmeonft networkas g rianp h-theocroenttieccx lte arclayr ries
certapienn altiebse naenfidt Tsh.ep enaltaireets h eh ugel ossienrs e levant
informattihoegn a;i ncso mei nt heh ighelre vela bosft racttihoern e,l ative
easwei thw hichl argneu mbers of cnoemtpwloerxck asn b eh andleadn d
compareadn,d i ng reatfleerx ibilFilteyx.i bisltietmyns o to nlyf rom
analogubeest weenne tworiknsp hysicaanld n on-physiscyaslt e(msse e
Chap5.. IIIb)u,tf romt hea bilittosy w itcphr oblefmrso mo ne' naturally­
occurrimnogd'et oa nothwehre rseo lutimoa1y1b se m oree asiolbyt ained.
Thusw e may translnaettew oriknst om atricaensd t apt hep owerful
resourocfem sa triaxl geb(rsae Ceh ap.1 .II(2o)r)p ,o rtrianyf ormation
ini nput-outmpauttr icaesns e twor(ksse Ceh ap.4 .II). Simwiel caarnl y
mover atheera siflryo mfl owp robletmosb arriperro blemasn,da ccom­
modatreo uteasn db oundarwiietsh icno mmon network f(oCrhmaapt.s
1. IIII)n.t hicsh aptetrh et opolosgtircu ctoufrt eh remea jorc lassoefs
networ(kFsi g1. .2 i)s e xamineAdn.a lysbiesg inwsi tthh er elatisviemlpyl e
8 BRANCHING NETWORKS I II
.
structoufrb er anchinnegt wor(kIs )a,n dm oveso n toc onsidgerra phs
withc ircu(iItIs) a,n df inalcleyl lunleatr(s III)F.i g.1 . 2 givefsa miliar
map examploefst hesteh reteo polocgliacs s(eTsa bl1e.1 ).
II.B RANCHING NETWORKS
T11fier smta inc lasosfg eographniectawlo rckosn sideirsed di stinguished
byi ttsr e..Ie i kset ructu(rFeisgI .. I-)A. Thesceo nsiosfst e tosf c onnected
linewsi thouatn yc ompleltoeo ps;t oipno logitcearlm inoltohgeyya re
'. ..c onnectgerda phwsi thout ci(rcOurie1t,9s 6'3 p,. 1 30)A.p arftr om
theiarn astomossiencgt io(nwsh icfho rmc loseldo opss)t ream-channel
systehmasv ea ltlh ep ropertoifte hse t opolotgriecee :. gt.h eyf ollotwh e
rule tthraetec so nsissitm ployfl inessu ccessiavdedleydt oe xistliinnge s,
sot haat t rewei thn vertihcaessn le dgeFsr.o mt het opologviiceawl­
-
points,t reanme tworks have the simplest poasnsdia bllle connectivity
-
•
'
Fig. T1o.p4o.l ogiciadlelnyt icchaaln nenle tworkAsr.r owheaidn dicaotuetsl et.
SourcSeh:r ev1e,9 66p,.2 8.
bifurcanteitnwgo rkssu,c ha sa rei llustrianFt iegd1. . 4a,r et opologically
similiarn;d eemda nyg eometricdailffleyr ent sntertewaomr akrse t opo­
logicaildleyn ti(cSahlr ev1e9,6 6..H) o wevert,h itso pologdiecfailn iotfi on
a treien cludfeosr ms intuirteicvoeglnyi zaabssl ien glpea thse:. gt.h e
singllien ceo nnectai nsge otf p ointTsh.i sp articuklianrdo ft opological
treken,o wna sa n' opeend get raipno's,e lso catiornaatlh tehra ns tructural
probleamnsd i sd iscusisneC dh ap4.. I(3H)e.r ed iscussiisco onn finteod
thes tructoufrb er anchinnegt worakssi llustrbaytt ehdeg raphosfs tream­
channesly stems.
Streanme twormkasy bev ieweadsp langer aph(sS cheideg1g9e6r7,A )
oft opologicalr oofintiettdre e e(sM eltoin9,5 9i)n,w hicnho deVs,,( made
up ofo utetri posr s ourcets;i, n nefro rkosr b ifurcatbi;oa nnsda, ter­
minarlo ootr o utlert)a, r ec onnectbeydl inkEs,,( exterei;oi rn,t eriio)r ,
sot haotn loyn el inekx isbtest weeann yt won odest,h eu pstreeanmd o f
I BRANCIINIENTGW ORK9S
.II
eachl ink either cownintehct twiono gt helri nkosr terminaitnia n g
I9 67,
sourc(eS hreve, p.I 7 8).T hiss tatemelneta dtso a problem of
definitiinot nh ats treanme tworiknsv olovnel yfl owsd irectferdo mt he
(9I6 A7, 04)
sourcteost heo utleatn,d S cheidegger p.l rejectthset erm
'treient' h aitt i mplineosn -direactrec(dse dgeHse) .p ropostehsed irected
term' arboresciennw chei'c hp endanvte rtiocrefi sn gertiepmsa nate from
a rootW.h atevetrh et erminolaodgoyp tetdh,e f ollowsinigm plree lation­
1959;
shipesx isbte tweetnh en umberosf nodesa nd link(sM elton.,
1967; 1967) 1.5):
Shreve, Woldenberg, (Fig.
E= Vb-rt
I
E=2Vt-I
Vb+ Vr= Vt
It is potsosd iebslcer itbheet opoloogfys treanme tworiknsa number
1967,
ofd ifferewnaty sS.h reve( pp.1 82-3h)a sp roposetdh es ymbolic
representaotfia o nne tworbky thef ollowipnrgo cedur'eS:t arattt he
Vt
p
I
I
I
I
I
I
-o
\ -- __
\
\
\
\
b
'\
'
'
'
'
't>
_..o
-­
_..,.
Fig. edg(eEsEi e,)
I.5I.d eaclh annenle tworskh owinign nearn do uter andr oot,
(VVrb,V, t ).
branchianngd t erminvaelr tices SourceM:e lton1,9 59.
outlaentd travtehresn ee tworakl,w aytsu rnilnegf attf orks raenvde rsing
directaitos no urceusn,t itlh eo utliesta gairne acheDdu.r intgh et raverse,
eneratae s equenocfei 'asn de 'bsy recordainn igt hefi rstt imea n
: teriolri niks t raversaendda ne thefi rstt imea giveenx teriloirn iks
__
-raversedE.a chl inwki lble traverstewdi cbeu tr ecordeodn lyo nce.'
Fig. 1s.h6o-wAss ucha symbolriecp resentawthiiocnhw, i leln sure that
opologiclayl identinceatlw orwkisl hla vei dentisceaqlu encBeesc.a use of
e relationship:
�
Ei== Ee-
I
)
- e numbero fe 'nse vere xceetdh ei 's( excepfto rt hefi nale , and by
+I 1,
aciniga nde equal toa nd- respectiav eglrya,p ohf p artisaulm s
1.6-B).
can beg enerat(eFdi g.
(9 67A,0 )
SimilarSlcyh,e ideggle r p.l 5 woulde xpretshsea rborescence
.6-A
�0,\7Il inF ig.I asa leftto r igh'tw ordT'h.i sw ordi sc omposeodf