On the Combinatorics of Rooted Binary Phylogenetic Trees

Transcription

1 O the Combiatoics of Rooted Biay Phylogeetic Tees Yu S. Sog Apil 3, 2003 AMS Subject Classificatio: 05C05, 92D15 Abstact We study subtee-pue-ad-egaft (SPR) opeatios o leaf-labelled ooted biay tees, also kow as ooted biay phylogeetic tees. This study is motivated by the poblem of gaphically epesetig evolutioay histoies of biological sequeces subject to ecombiatio. We ivestigate some basic popeties of the iduced SPR-metic o the space of leaf-labelled ooted biay tees with leaves. I cotast to the case of uooted tees, the umbe U T of tees i which ae oe SPR opeatio away fom a give tee T depeds o the topology of T. I this pape, we costuct ecusio elatios which allow oe to detemie the uit-eighbouhood size U T efficietly fo ay tee topology. I fact, usig the ecusio elatios we ae able to deive a simple closed-fom fomula fo the uit-eighbouhood size. As a coollay, we costuct shap uppe ad lowe bouds o the size of uit-eighbouhoods ad ivestigate the diamete of. Lastly, we coside a eumeatio poblem elevat to populatio geetics. Keywods: ooted tees, odeed tees, subtee pue egaft, eighbouhood 1. Itoductio Biology abouds with examples whee gaphical epesetatios ad combiatoics have poved vey useful. Though this bidge betwee biology ad mathematics, may iteestig ideas have bee caied ove fom the latte ad have lead to sigificat developmets i the fome. Of paticula iteest to geeticists is the usage of tees to epeset evolutioay histoies of biological sequeces. I additio to obtaiig a tee which best descibes the evolutioay elatioship of give sequeces, oe is ofte also iteested i kowig how diffeet a tee is fom othe tees; that is, oe is iteested i a quatitative measue of how fa a tee is fom aothe. The aswe to that questio, of couse, depeds o how the distace is measued, ad theefoe oe eeds to specify which metic should be used i measuig the distace betwee two tees. A type of metic widely used i biology is that defied i tems of cetai opeatios which eaage tees [1, 5]; the distace betwee two tees is defied as the miimum Depatmet of Statistics, Uivesity of Oxfod, 1 South Paks Road, Oxfod, OX1 3TG, UK, sog@stats.ox.ac.uk 1

2 umbe of opeatios equied to tasfom oe tee to the othe. A paticula kid of opeatio which will be the focus of this pape is the so-called subtee puig ad egaftig [5]. I a subtee-pue-ad-egaft (SPR) opeatio, oe detaches a edge fom a tee T, thus puig a subtee t fom T, ad egafts t to somewhee else o the emaiig pat of T. We defe a moe pecise defiitio of SPR opeatios util 2. I [1], Alle ad Steel have cosideed the space u of leaf-labelled -leaved u uooted biay tees. Afte havig defied the eighbouhood of a tee T as the set of all tees i u which ae oe SPR opeatio away fom T, Alle ad Steel have show that the size of the eighbouhood of T does ot deped o the topology of T ad is equal to Moeove, they have show that the diamete diam SPR, measued usig the SPR-metic, satisfies the followig bouds [1]: u 2 o diam SPR u 3 (1.1) I the peset pape, we ivestigate aalogous questios fo ooted tees, which, as we discuss pesetly, ae moe elevat to biology tha uooted tees. I cotast to the case of uooted tees, the size U T of the eighbouhood of a leaf-labelled ooted biay tee T depeds o the topology of T. We ae, howeve, able to costuct ecusio elatios which ca be used to compute U T efficietly fo ay tee topology type. Futhemoe, usig the ecusio elatios, we deive a simple closed-fom fomula fo U T. We fid two paticula topology types, oe of which ealises the maximum value of U T ad the othe the miimum, ad we combie this fidig with the afoemetioed esults to costuct shap bouds fo U T. Also, we show that the diamete of satisfies bouds simila to that show i (1.1). Whe epesetig geealogical pocesses by tees, it is atual to use ooted tees istead of uooted tees, fo the existece of a distiguished poit o a tee eables us to defie a sese of time diectio; that is, time flows fom the oot to the leaves. This distictio betwee ooted ad uooted tees leads to obsevable diffeeces i pactice. Fo istace, i [2] Hei has poposed a algoithm fo ecostuctig the most pasimoious evolutioay histoies of sequeces which have udegoe ecombiatio. As he poits out i the pape, if uooted tees ae used i the algoithm, iteal cotadictios might aise, thus pevetig the costuctio of a gaphical epesetatio. If ooted tees ae used, howeve, it could be possible to compute the exact miimum umbe of ecombiatio evets ad theeby costuct a cosistet gaphical epesetatio [4]. Ou fidigs fom the peset pape ae used i [4], whee SPR opeatios o ooted tees ae used to epeset ecombiatio evets. I geetics, oe is atually lead to coside leaf-labelled ooted biay tees whose iteal vetices, which coespod to biological evets, ae totally odeed. Such tees ae sometimes called odeed tees. I this pape we coside a eumeatio poblem which aises i populatio geetics. Namely, we deive closed-fom fomulae fo the umbe of ooted ad odeed tees which ae compatible with a bipatitio of the label set. This pape is ogaised as follows. I 2 we lay out some basic defiitios ad state a few fudametal esults egadig the combiatoics of leaf-labelled ooted biay tees. I 3 we costuct ecusio elatios fo the size of uit-eighbouhoods ad deive a closed-fom fomula fo U T. I 4 we obtai shap uppe ad lowe bouds 2

3 o the size of uit-eighbouhoods. The diamete of the space is discussed i 5. I 6 we discuss the afoemetioed eumeatio poblem elevat to geetics. (NOTE: We have witte a compute pogam to check explicitly all ou esults fo 9.) 2. Pelimiaies 2.1. Defiitios By a ooted biay phylogeetic tee we mea a leaf-labelled ooted biay tee whose bach legths ae ot specified. The space of leaf-labelled ooted biay tees with leaves is deoted by. The degee of a vetex v is the umbe of edges which ae icidet with v. Fo 2, a tee i has labelled degee-1 vetices called leaves; 2 ulabelled degee-3 vetices; ad a distiguished vetex of degee 2 called the oot. A 1-leaved tee cosists of a sigle labelled degee-0 vetex which seves as both the oot ad the leaf. A vetex which is ot a leaf is called a iteal vetex. The leaves of a -leaved tee ae bijectively labelled by a fiite set L of elemets. Let L T be the label set fo the leaves i T. The, fo a subtee s T, L s L T deotes the label set fo the leaves i s. I the emaide of this pape, whe we say a tee without ay qualificatio, we shall mea a leaf-labelled ooted biay tee. We say that two vetices u v T ae adjacet if thee exists a edge which jois u ad v. A path fom vetex v 0 to vetex v k is a alteatig sequece v 0 e 1 v 1 e 2 e k v k of vetices v i ad edges e i, such that (1) e i jois v i 1 ad v i, ad (2) all e i s ad v i s ae distict. Fo v a degee-3 vetex i T, we defie γ v as the umbe of degee-3 vetices, ot icludig v itself, i the path betwee v ad the oot of T. I a ooted tee, time flows fom the oot to the leaves. We say that vetex v T is a descedat of vetex u T if thee exists a path fom u to v which goes stictly fowad i time; u is called a acesto of v. A subtee s of a tee T is a tee i, whee, ad is defied by the popety that if a vetex v T is cotaied i s, the so ae all its descedats. I this pape a subtee whose oot is adjacet to the oot of T is called a R-subtee. A -leaved ooted biay tee cotais 2 2 edges. Fo ay (sub)tee s, we deote by s the umbe of leaves i s ad defie η s : 2 s 2, which is equal to the umbe of edges i s. Lemma 2.1. (Schöde) The umbe of iequivalet leaf-labelled ooted biay tees with leaves is [3] R : 2 3!! ! 2 1 1! 2.2. SPR Opeatios Thee ae thee kids of SPR opeatios that ca be pefomed o leaf-labelled ooted biay tees. A illustatio of these opeatios is show i Figue 1. I what follows, let T (esp. T ) deote a tee befoe (esp. afte) a SPR opeatio. The otatio T t 3

4 deotes the pat of T obtaied fom emovig a subtee t ad the edge icidet with the oot of t but ot cotaied i t. I wods the thee SPR opeatios ae as follows. (1) A edge e is cut to pue a o-r-subtee t, ad t is egafted oto a pe-existig edge i the emaiig pat T t of T, thus ceatig a ew degee-3 vetex. The vetex i T t whee e used to be icidet gets emoved. The oot of T emais the oot of T. (I Figue 1, T T 1 is a example of this kid. The edge e b is cut ad the egafted oto the edge e a.) (2) Let s 1 ad s 2 be the two R-subtees of T, ad let e 1 ad e 2, espectively, be the edges which joi thei oots to the oot of T. The edge e 1 is cut to pue s 1, ad s 1 is egafted oto a pe-existig edge i s 2. The edge e 2 gets emoved ad the degee-3 vetex i s 2 whee e 2 used to be icidet gets eplaced by a degee-2 vetex, which becomes the oot of T. (I Figue 1, T T 2 is a example of this kid. The edge e c is cut ad egafted oto e a. The oot of the R-subtee cotaiig t 1 t 2 ad t 3 becomes the oot of T 2.) (3) A edge e is cut to pue a o-r-subtee t, ad t is joied to the oot of T. The oot of T is give by ceatig a ew vetex of degee 2 o e. (I Figue 1, T T 3 is a example of this kid. The edge e b is cut ad the joied to the oot of T. A ew degee-2 vetex is ceated o the edge ad it seves as the oot of T 3.) Root e a e b e c t 1 t 2 t 3 t 4 t 5 T Root 1 SPR 1 SPR Root 1 SPR Root t 1 t 2 t 3 t 4 t 5 t 1 t 4 t 5 t 2 t 3 t 1 t 3 t 4 t 5 t 2 T 1 T 2 T 3 Figue 1: A illustatio of SPR opeatios. Big ope cicles labelled by t j epeset subtees. Fo ay pai of tees T T, we measue the distace betwee them usig the SPR-metic d : 0; that is, the distace d T T is a o-egative itege defied as the miimum umbe of SPR opeatios ecessay to tasfom T ito T. 4

5 3. The Uit-Neighbouhood of a Tee We defie the uit-eighbouhood of a tee T as U T T d T T Topology Types We hee defie two topology types, show i Figue 2, fo which simple ecusio elatios fo U T will late be fomulated. A type A tee is chaacteised by the featue that oly a sigle leaf is o oe side of the oot. I Figue 2(b), if v deotes the degee-3 vetex with which the leaf l is adjacet, the k γ v. The otio of left ad ight i the figue is ielevat. Fo ease of efeece, we have give labels to some edges; these labels ae ot a pat of the defiitio of a leaf-labelled ooted biay tee. The eade should efe to the captios theei fo futhe explaatio. We emphasise that what we itoduce hee does ot defie a classificatio, sice a tee ca fall ito moe tha oe type. Fo istace, fo 3, ay tee T is type B, but it may also be type A. Root Root e a e e d e b c t e L e 1 e 2 t 1 t 2 s 1 s 2 s k 1 s k l l li TYPE B k 0 e d e k e b e c e a TYPE A (a) (b) Figue 2: A schematic epesetatio of topology types. Big ope cicles ad boxes epeset subtees, ad evey subtee cotais at least oe leaf. Leaves ae labelled by l l i L. (a) A -leaved type A tee. (b) A -leaved type B tee. e k Opeatios o Tees We hee defie a eductio opeatio which will be used i ou ecusio elatios. The opeatio we pesetly defie educes the umbe of leaves i a tee by oe. I the followig discussio, we use the labels show i Figue 2. (1) If T is type A, the T l is give by emovig the leaf l ; emovig the oot of T ; emovig the edges e a ad e d ; ad makig the vetex whee e a, e b, e c used to be icidet ito the oot of T l. A example of this kid of opeatio is illustated i Figue 3(a). 5

6 (2) If T is type B, the T l is give by emovig the leaf l ; emovig the edge e c ; emovig the degee-3 vetex whee e a, e b, e c used to be icidet; ad megig the edges e a ad e b ito oe. A example of this kid of opeatio is illustated i Figue 3(b). l l t 1 t 2 t t 2 t 1 t 2 t 1 t l 1 2 l l i T (a) T l T (b) T l Figue 3: A illustatio of opeatios. (a) T is type A. (b) T is type B, with k 1. l i 3.3. Recusio Relatios fo U T I this subsectio, we costuct ecusio elatios fo U T. The topology types defied i 3.1 costitute a athe coase desciptio. Fo istace, type B ecompasses may distict tee topologies. It is iteestig to ote that the depedece of type B ecusio elatio o tee topology is ecoded i a sigle paamete k. The ecusio elatios ca be applied i seveal diffeet ways, depedig o which type oe chooses to call a tee ad which opeatio oe chooses to use. The fial aswe fo U T, howeve, does ot deped o how oe chooses to compute it. I fact, type B ecusio elatio aloe is sufficiet fo computig U T fo ay tee T. Type A ecusio elatio, howeve, will be useful fo ou discussio i 4. Popositio 3.2. Let 4 ad let T. The, the size of the uit-eighbouhood U T satisfies the ecusio elatio U T l 6 16 if T is type A U T U T l 2 4 k 11 if T is type B whee l is as show i Figue 2, ad T l is a 1 -leaved tee obtaied usig the opeatio defied i 3.2. I the ecusio elatio fo type B tees, k is a o-egative itege defied as i Figue 2(b). REMARK: U T 2, fo all T 3, seves as the bouday coditio fo the ecusio elatios. Poof. We have divided ou poof ito seveal pats. I ou discussio, we shall cofom to the otatios show i Figue 2. TYPE A: (A-1) A tee i U T ca be geeated by a sigle SPR opeatio withi the pat to the left of the oot of T. Thee ae U T l such opeatios. 6

7 (A-2) Ay edge except fo e a ad e d ca be detached fom T ad egafted oto e d to geeate a tee i U T which has ot bee icluded i (A-1). Thee ae η T such edges i T. (A-3) The edge e d ca be detached ad egafted oto a edge othe tha e a e b e c ad e d to geeate a tee i U T which has ot bee icluded i (A-1) o (A-2). Thee ae η T possibilities. (A-4) Ay edge except fo e a e b e c ad e d ca be detached fom T ad egafted oto the oot of T to geeate a tee i U T which has ot bee icluded above. Thee ae η T such edges. Addig up the cotibutios gives U T U T l U T l 6 16 TYPE B (k 1): (B-1) Thee ae U T l iequivalet SPR opeatios which do ot diectly ivolve e a o e c (i.e. eithe cuttig them o egaftig oto them). (B-2) The edge e a ca be detached fom T ad egafted oto ay edge except fo e a e b ad e c to yield a ew tee i U T. Thee ae η T iequivalet such SPR opeatios. (B-3) The edge e c ca be detached fom T ad egafted oto ay edge except fo e a e b e c e d e k to geeate a tee i U T which has ot aleady bee accouted fo i (B-1) o (B-2). Thee ae such SPR opeatios. (B-4) Ay edge i the subtee t ca be detached fom t ad egafted oto eithe e a o e c. Thee ae 2η t such opeatios which geeate iequivalet tees i U T. (B-5) Fo j 1 2 k 1, if s j 1, ay edge i the subtee s j ca be detached fom s j ad egafted oto eithe e a o e c. Also, e j ca be detached fom T ad egafted oto eithe e a o e c. Thee ae a total of 2 k j 1 1 η s j 1 iequivalet such SPR opeatios. (B-6) If s k 1, ay edge i the subtee s k ca be detached fom s k ad egafted oto eithe e a o e c. Thee ae 2η s k such SPR opeatios. (Note: detachig e k ad egaftig it oto eithe e a o e c geeates a tee aleady icluded i (B-2).) (B-7) The edge e a o e c ca be detached ad egafted oto the oot. This cotibutes 2 to U T. (B-8) The edge e L ca be detached ad egafted oto eithe e a o e c to yield a tee i U T which has ot aleady bee icluded above. This cotibutes 2 to U T. 7

8 I summay, we have U T U T l η t 2 η s j 1 2η s k 4 j 1 k U T l η t η s j k 1 2 U T l 2 4 k 11 whee the last lie follows fom η t k j 1 η s j 2 2k 6. TYPE B (k 0): I this case, the oly edges to the ight of the oot ae e a e b ad e c (c.f. Figue 2(b)). j 1 k 1 (B-1 ) Thee ae U T e c diectly. l iequivalet SPR opeatios which do ot ivolve e a o (B-2 ) A edge i the subtee t ca be detached fom t ad egafted oto eithe e a o e c. Thee ae 2η t iequivalet such opeatios. (B-3 ) The edge e c ca be detached fom T ad egafted oto ay edge to the left of the oot. Thee ae η T such opeatios. (B-4 ) Likewise, the edge e a ca be detached fom T ad egafted oto ay edge to the left of the oot. Agai, thee ae η T such opeatios. Note that, sice k 0, detachig e a (esp. e c ) ad egaftig it to the oot of T is equivalet to detachig e c (esp. e a ) ad egaftig it to e L. Also, detachig e L ad egaftig it to e a (esp. e c ) is equivalet to detachig e a (esp. e c ) ad egaftig it to e L. These opeatios geeate tees which have aleady bee icluded i the above list. Hece, we obtai U T U T l 2η t U T l whee η t has bee used A Closed-Fom Fomula fo U T As we have metioed befoe, it is always possible to compute U T oly usig the type B ecusio elatio. Applyig the ecusio elatio i a systematic way, oe ca obtai the followig esult: Popositio 3.3. Let 3 ad let T. Let v 1 v 2 v 2 be the set of degee-3 vetices i T. The, with γ v i defied as i 2.1, U T is give by 2 U T γ v i (3.2) i 1 8

9 Poof. Label the degee-3 vetices of T by v 1 v 2 v 2 so that γ v 2 γ v 3 γ v 1 (3.3) Now, successively pefom opeatios i the ode l 1 l 2 l 3 (3.4) whee l i deotes a leaf icidet with v i i T i k 1 1 l k. Note that, because of the imposed odeig i (3.3), l i opeatio does ot chage the value of γ v j fo j i 1 i 2 2. Afte pefomig all the opeatios i (3.4), we ed up with a 3-leaved tee, whose uit-eighbouhood size is 2. I summay, usig Popositio 3.2 we obtai 3 2 U T 2 4m 11 2 γ v i γ v i m 42 i 1 whee i the last equality γ v 2 has bee added to the summatio (Note that the odeig i (3.3) implies that γ v 2 0, so addig it to the summatio does ot chage the value of U T ). Recall Alle ad Steel s fomula [1] AS : fo the size of the uit-eighbouhood of a uooted biay tee i. The fist tem i ou fomula (3.2) is oe othe tha AS 1. This esult eflects the fact that thee exists a oe-to-oe coespodece betwee the set u 1 of leaf-labelled uooted biay tees with 1 leaves ad the set of leaf-labelled ooted biay tees with leaves. Futhemoe, the defiitio of a SPR opeatio fo -leaved ooted tees is moe estictive tha that fo 1 -leaved uooted tees. That is, thee ae moe iequivalet SPR opeatios fo 1 -leaved uooted tees tha fo -leaved ooted tees. The secod tem i (3.2) is the ecessay coectio tem which accouts fo this fact. 4. Shap Bouds o the Size of Uit-Neighbouhoods I this sectio, we defie two special types of tees ad examie the size of thei uiteighbouhoods. We the use ou fidigs to obtai shap uppe ad lowe bouds fo U T Two Special Types of Tees Coside the sequece a 1 a 2 a 3 whee a m i 1 log 2 m 1 1 (4.5) Hee, deotes the geatest itege fuctio, also kow as the floo fuctio. Moe explicitly, the sequece is of the fom u 9

10 cotaiig two 0s, fou 1s, eight 2s, sixtee 3s, thity two 4s, etc. Also, coside the stictly ascedig sequece b 1 b 2 b 3 whee b m m 1; that is, the sequece is cotaiig oe 0, oe 1, oe 2, oe 3, oe 4, etc. Let T. If its 2 degee-3 vetices ca be labelled by v 1 v 2 v 2 so that γ v m a m (esp. γ v m b m ) fo evey m 1 2 2, the we shall call T a γ-expoetial (esp. γ-uifom ) tee. Examples of γ-expoetial ad γ-uifom tees ae show i Figue 4. (SIDE REMARK: The ame γ-expoetial is deived fom the fact that o-egative iteges ae expoetially distibuted i a 1 a 2, ad the ame γ-uifom fom the fact that o-egative iteges ae uifomly distibuted i b 1 b 2.) (a) Figue 4: Examples of special types of tees. Leaf labels ae suppessed ad the values of γ v i fo degee-3 vetices v i ae show. (a) A 10-leaved γ-expoetial tee. (b) A 10-leaved γ-uifom tee. (b) 4.2. Uit-Neighbouhoods of γ-expoetial ad γ-uifom Tees I this subsectio, we examie the size of uit-eighbouhoods of γ-expoetial ad γ-uifom tees. As we show i the followig popositio, these tees ae special i the sese that they ealise exteme values of the uit-eighbouhood size. Popositio 4.4. Let 4 ad defie δ max : max T U T ad δ mi : mi T U T. The, I. U T δ max if ad oly if T is γ-expoetial, II. U T δ mi if ad oly if T is γ-uifom. Poof. Coside the case 4. It tus out that evey tee T 4 is eithe γ-expoetial o γ-uifom. Moeove, we ca use Popositio 3.2 o Popositio 3.3 to show explicitly that U T 12 if T is γ-expoetial, wheeas U T 10 if T is γ-uifom. Hece, both statemets I ad II i the popositio ae tue fo 4. PROOF OF PART I: Let H I deote the iductio hypothesis that, fo 4 1 whee 5, U T δ max if T is γ-expoetial. Let T be a -leaved γ-expoetial tee ad let l deote a leaf adjacet to a iteal vetex v with γ v a 2, whee a 2 is defied i (4.5). The, T l also is a γ-expoetial tee. Hece, by ou iductio 10

11 hypothesis H I, we kow that U T l δ max 1. Moeove, sice a γ-expoetial tee is type B fo 4, we ca use the ecusio elatio fom Popositio 3.2 to obtai U T δ max a 2 11 Suppose T is type A. The, U T l δ max 1 whee l is the sigle leaf o the ight had side of the oot i Figue 2(a). Futhemoe, sice a 2 11, fo all 5, we coclude that U T U T l 6 16 U T. We ow show that fo all type B tee T which is ot γ-expoetial, U T U T. Let l be a leaf i T such that k is as lage as it ca be i the followig fomula fom Popositio 3.2: U T U T l 2 4 k 11. By defiitio, δ max 1 U T l. Futhemoe, sice T is ot γ-expoetial, k a 2 fo all 5, ad theefoe 2 4 a k 11. Hece, U T U T, ad we thus coclude that if T is γ-expoetial, the U T δ max. This completes ou iductio. The covese ca be show as follows. Let T be a -leaved tee such that U T δ max. The, fom the fomula fo U T i (3.2), we kow that 2 i 1 γ v i must be as small as possible. But, i a ooted biay tee, γ v 1 γ v 2 γ v 2 a 1 a 2 a 2 gives the miimum value of 2 i 1 γ v i, ad theefoe T must be γ-expoetial. PROOF OF PART II: Assume that, fo 4 1 whee 5, U T δ mi if T is γ-uifom. We efe to this assumptio as hypothesis H II. Let T be a -leaved γ-uifom tee. Sice a γ-uifom tee is of type A, we ca use the type A ecusio elatio fom Popositio 3.2 to obtai U T U T l 6 16 δ mi whee the secod equality follows fom the iductio hypothesis H II. Suppose T is type B. The, U T U T l 2 4 k 11 δ mi k 11, whee k is as show i Figue 2(b). But, i a tee with leaves, k is bouded fom above. Moe pecisely, k 3, ad we theefoe have k 11 fo evey 5. Hece, we coclude that U T U T. Suppose T is a type A tee which is ot γ-uifom ad let l deote the sigle leaf o the ight had side of the oot i Figue 2(a). The, applyig the type A ecusio elatio gives U T U T l 6 16 δ mi Hece, U T U T. We have thus show that if T is a -leaved γ-uifom tee, the U T δ mi. This completes ou iductio. We ow sketch the poof of the covese. Let T be a -leaved tee such that U T δ mi. The, it implies that i 1 2 γ v i must be as lage as possible i (3.2). It is easy to show that, i a ooted biay tee, γ v 1 γ v 2 γ v yields the maximum value of i 1 2 γ v i. Thus, T must be γ-uifom The Bouds Usig the ecusio elatios fom 3.3 ad the esults fom 4.2, we ca deive a coollay of the followig fom: 11

12 Coollay 4.5. Fo all T, the bouds 4, the size of the uit-eighbouhoodu T satisfies U T m 1 log 2 m 1 That is, δ mi ad δ max m 1 log 2 m 1. Poof. Let T be a -leaved γ-uifom tee. The, by Popositio 3.3 ad Popositio 4.4, we have 2 U T δ mi b m m 1 whee b m m 1 (c.f. 4.1). Hece, δ mi Let T be a -leaved γ-expoetial tee. The, it follows fom Popositio 3.3 ad Popositio 4.4 that 2 U T δ max a m log 2 m 1 m 1 m 1 ad we have ou desied esult. 5. Diamete of As Alle ad Steel have doe fo uooted tees [1], we ca obtai the followig esult fo ooted tees: Popositio 5.6. Let 3 ad let diam SPR ( ) deote the diamete of the maximum value of d T T ove all tees T T. The,, defied as 2 o diam SPR ( ) 2 Poof. Fom Coollay 4.5, we kow that δ max Followig exactly the same lie of easoig as Alla ad Steel have doe i [1], oe ca aalyse diam SPR 2 3!! usig Stilig s appoximatio to deive the lowe boud. The lowe boud fo the ooted case is the same as that i the uooted case, because fo both cases the uiteighbouhood size gows quadatically with espect to. Fo small values of, say 6, it is easy to come up with examples of T T such that d T T 2. We wish to show that, fo all 3, 2 SPR opeatios ae sufficiet to tasfom ay T 1 to ay T 2. Note that the oot of a tee patitios the label set L ito two disjoit pope subsets of L. Let A 1 A c 1 ad A 2 A c 2 be such bipatitios of L associated to two tees T 1 ad T 2, espectively. Hee, A i deotes a 12

13 pope subset of L ad A c i its complemet elative to L. Let S 1 A 1 A c 1 ad S 2 A 2 A c 2. Sice A 1 A c 1 A 2 ad A c 2 ae pope subsets, if S 1 S 2, the S1 c S 2 ad S 1 S2 c. Theefoe, it is always possible to label the bipatitios so that A 1 A 2 ad A c 1 A c 2. Upo makig such a choice of labellig, let l i A 1 A 2 ad l j A c 1 A c 2. Note that the leaves l i ad l j ae o opposite sides of the oot i both T 1 ad T 2. Now, pue all the leaves, except fo l i ad l j, fom T 1 ad the egaft those 2 leaves, labelled by L l i l j, to make T 2. It is clea that this is always possible. Thus we coclude that d T 1 T 2 2 fo all T 1 T Numbe of Tees Compatible with a Bipatitio of L The set v 1 v 2 v 2 of degee-3 vetices i T is a patially odeed set whose biay elatio deoted is give by acestal elatio; we say that v i v j if v i is a descedat of v j. Two degee-3 vetices v i ad v j ae icompaable if v i is ot i the path to the oot fom v j ad vice vesa. A odeed tee is a leaf-labelled ooted biay tee whose coespodig set v 1 v 2 v 2 of degee-3 vetices is a totally odeed set; that is, fo ay two vetices v i ad v j, eithe v i v j o v j v i. I this case, the biay elatio is give by age odeig. As befoe, v i v j if v i is a descedat of v j. If thee exists o acestal elatio betwee v i ad v j, the eithe v i v j o v j v i is allowed. Futhemoe, we impose the coditio that v i v j if i j. Two tees equivalet as leaf-labelled ooted biay tees ae distict as odeed tees if the odeig of thei degee-3 vetices ae diffeet. It is well-kow i populatio geetics that the umbe of iequivalet odeed tees with leaves is D : m 2 Recall that R : 2 3!!. m 2! 1! 2 1 Let B B c deote a bipatitio of the label set L ito two pope subsets. A tee T is said to be compatible with the bipatitio B B c if thee exists a edge i T such that cuttig the edge decomposes T ito two coected compoets, oe cotaiig the leaves labelled by B ad the othe the leaves labelled by B c. I populatio geetics fo example, whe usig the so-called ifiite-sites model the umbe of tees compatible with a bipatitio of L is a quatity of iteest. Suppose B k ad B c k, ad let w k (esp. w o k ) deote the umbe of ooted tees (esp. odeed tees) compatible with the bipatitio B B c. Clealy, if k 1 o k 1, the w k R ad w o k D. Fo 2 k 2, it is ot difficult to show that the umbe of ooted tees compatible with B B c is w k : 2 3 R k R k Fo odeed tees, we have the followig esult: Popositio 6.7. Fo 4 ad 2 k 2, the umbe of odeed tees compatible with the bipatitio B B c, whee B k ad B c k, is w o 2 k : D k D k k 1 k 1 k 1 13

14 p k 2 p Poof. We fist show that w o k is give by the followig expessio: k w o 1 p 1 k s k : D k D k p 1 p 0 s 0 2 k 1 p 1 D k D k p p k 2 k s 1 p 0 p s 0 2 D k D k 2 k 1 (6.6) Coside a u cotaiig k black balls labelled by B ad k white balls labelled by B c. Daw two balls fom the u. If oe black ball ad oe white ball ae daw, the simply eplace both balls back ito the u. If two black (esp. white) balls labelled X i ad X j ae daw, the eplace with a sigle black (esp. white) ball labelled X i X j. Hee, X i could be l i1 l i2 l i j, whee l i1 l i2 l i j L. Note that, i total, k 1 pais of black balls ca be daw. Whe the k 1 th pai of black balls labelled X i ad X j ae daw, the eplace with a white ball labelled X i X j. If oly white balls emai i the u, keep dawig pais ad eplace with a white ball with a ew label util oly oe white ball emais i the u. Evey possible sequece of daws eds up with a sigle white ball labelled l 1 l 2 l, whee l 1 l 2 l L. Thee exists a oe-to-oe coespodece betwee the set of sequeces of distict u cotets which aise i the above u model ad the set of -leaved odeed tees which cotai a odeed subtee with k leaves labelled by B. The odeig of u cotets i a sequece coespods to the odeig of iteal vetices i a odeed tee. The iitial set of balls coespod to the leaves ad a ball with a composite label l i1 l i2 l i j coespods to a iteal vetex whose descedat leaves ae pecisely l i1 l i2 l i j. The white ball eplaced ito the u whe the k 1 th pai of black balls ae daw coespods to the oot of the k-leaved odeed subtee whose leaves ae labelled by B. Whe the k 1 th pai of black balls ae daw, we ca associate to the completed sequece of black ball cotets a odeed tee with k leaves labelled by B. By aalogy, thee ae D k iequivalet ways of achievig this. Suppose p pais of white balls have bee daw befoe the k 1 th pai of black balls ae daw. Sice all balls ae labelled, thee ae p s 1 k s 0 2 iequivalet ways of dawig p pais of white balls. Moeove, a sequece α 1 α 2 α p of p pais of white balls daw ad a sequece β 1 β 2 β k 2 p k 2 of k 2 pais of black balls daw ca be combied i p iequivalet ways to fom a loge sequece of legth p k 2 while maitaiig the odeig of α i s ad that of β i s. Recall that, whe the k 1 th pai of black balls ae daw, a labelled white ball is eplaced ito the u. Hece, the umbe of white balls the emaiig i the u is k p 1. Subsequet dawigs of pais of white balls with eplacemet coespod to geeatig D k p 1 odeed tees. Lastly, we ote that p ca take ay value betwee 0 ad k 1, iclusive. We have thus deived the expessio i the fist lie of (6.6). The expessio i the secod lie of (6.6) is obtaied by eplacig k with k, ad vice vesa, i the above paagaphs. It coespods to the umbe of -leaved odeed tees which cotai a odeed subtee with k leaves labelled by B c. 14

15 The last lie i (6.6) coects fo double coutig. That is, both the fist lie ad the secod lie i (6.6) iclude the umbe of odeed tees which cotai both a odeed subtee with k leaves labelled by B ad a odeed subtee with k leaves labelled by B c 2. The combiatoial facto k 1 is fo the odeig of the k 1 iteal vetices of a k -leaved odeed tee elative to the k 1 iteal vetices of a k-leaved odeed tee. This completes the deivatio of (6.6). Now, otice that s j i s 0 2 D i D i j 1, which implies that the expessio iside the backet i the fist lie of (6.6) is equal to D k k p 1 p k 2 2 p. Futhemoe, sice k 1 k p 1 p k 2 p 0 2 p k 1, the fist lie i (6.6) becomes D k D k k 1. I a simila vei, the secod lie of (6.6) ca be e-witte as D k D k k 1. This completes ou poof of the popositio. Ackowledgmets The autho gatefully ackowledges Jotu Hei ad Mike Steel fo useful commets o the mauscipt. This eseach is suppoted by EPSRC ude gat HAMJW, by MRC ude gat HAMKA, ad by a gat fom the Daish Natual Sciece Foudatio (SNF ). We ackowledge Oxfod Supecomputig Cete fo allowig us to use thei CPU time. Refeeces 1. B.L. Alle ad M. Steel, Subtee Tasfe Opeatios ad Thei Iduced Metics o Evolutioay Tees, A. Comb. 5 (2001) J. Hei, A Heuistic Method to Recostuct the Histoy of Sequeces Subject to Recombiatio, J. Mol. Evol. 36 (1993) E. Schöde, Vie Combiatoische Pobleme, Zeit. fü. Math. Phys. 15 (1870) Y.S. Sog ad J. Hei, Pasimoious Recostuctio of Sequece Evolutio ad Haplotype Blocks: Fidig the Miimum Numbe of Recombiatio Evets, to appea i: Lectue Notes i Compute Sciece, Poceedigs of Wokshop o Algoithms i Bioifomatics 2003, Spige Velag. 5. D.L. Swoffod ad G.J. Olse, Phylogey Recostuctio, i: Molecula Systematics, D.M. Hillis et al., Eds., Siaue Associates, Massachusetts, 1990, pp