Exact enumeration of cherries and pitchforks in ranked trees under the coalescent model

Exact en umeration of c herries and pitc hfork s in rank ed trees under the coalescen t mo del Filipp o Disan to ∗ Thomas Wie he ∗ April 16, 2018 Abstract W e cons ider exact enumerations and pr obabilistic prop erties o f r anke d trees when generated under the rando m coalescent process . Using a new approach (see [9, 10]), based o n generating functions, we deriv e several statistics s uch as the exact pr obability of finding k cherries in a r a nked tree of fix e d size n . W e then extend our metho d to consider also the num b er of pitchforks . W e find a recursive formula to calculate the joint and conditional probabilities of cherries and pitch- forks when the size of the tree is fixed. 1 In tro d uction Giv en a direction b y time, ancestry relationship b et wee n sp ecies, individuals, alleles or cells can b e d epicted as a ro oted tree. Of p articular in terest are binary ro oted un ordered trees. These can b e fu r ther classified into several sub classes. Here w e will r anke d tr e e s , whic h are defined b elo w . W e assu m e that trees are generated by the coalesce n t pro cess. An imp ortant parameter is the num b er of cherries of a tree. By a new approac h based on generating functions w e extend previous results (see for example [9]) deriving an exact f orm ula for the probab ility of find ing k c her- ries in a rank ed tree of size n . F urthermore, we sh o w that sev eral kno wn statistics (see [10]) concerning pitchforks follo w as corollaries from a partial differen tial equation whic h also giv es an efficie n t recursion to compute the conditional probabilit y d istr ibution of p itc hforks given a certain num b er of c herr ies. ∗ Institut f¨ ur Genetik, Universi t¨ at zu K¨ oln; Z ¨ ulpic her Straße 47a, 50674 K¨ oln, German y 1 One motiv ation for this study co mes from p opulation genetics and the question ho w ’t ypical’ c o alesc ent tr e es [13] lo ok lik e. Our results giv e some insigh t in to stru ctural prop erties of trees generated under th e standard n eu- tral mo del [12]. These r esults provide a reference against wh ic h non n eu tral and/or n on ind ep enden tly generated trees may b e compared. T o illustr ate the latter w e pay atten tion to trees whic h are linke d along a recom binin g c hromosome. 2 Preliminaries W e s tart with some basic d efinitions. A binary r o ote d tree is a tree with a ro ot and in wh ic h all n o des h a v e ou td egree either 0 or 2. No d es with out- degree 2 are called i nternal , no des with outdegree 0 are external . External no des are also called le aves . T he size n of a tree is the num b er of its external no des. The subtr e e of an in ternal no de i is the tree with ro ot i . A tree is said to b e un-or der e d wh en it is taken in th e graph theoretic sense so that subtrees stemming f rom an inte rnal no de ha v e not a left-right order b etw een themselv es. Here, we care ab out tree top ology and we d o not care ab out branc h lengths. W e consider the follo wing class. A binary un-ord er ed tree of size n is said to b e a r anke d tr e e if the set of inte rnal no des is totally ordered b y lab e ls b elonging to { 1 , 2 , ..., n } in suc h a wa y that eac h child’s lab el is greater than its paren t’s lab el, (see Fig. 1). The total order of internal lab els can b e in terpreted as a historical time order; accordingly , Hardin g [4 ] calls suc h trees histories . W e will denote b y R the set of rank ed trees and by R n the set of trees of size n . In wh at follo w s, n = n ( t ) alw a ys repr esen ts the num b er of lea ve s of a ranked tree t . The cardin alit y of the set R n is giv en by the follo wing exp onential gen- erating function R ( x ) = X n ≥ 0 |R n | n ! x n = sec( x ) + tan( x ) . (1) whose first co efficien ts |R n | (with n > 0) are 1 , 1 , 1 , 2 , 5 , 16 , 61 , 272 , .... Rank ed trees can b e bijectiv ely mapp ed to 0-1-2- incr e asing tr e es (see Callan, 2005; http:/ /www.stat .wisc.edu/ ~ callan/n otes ). F rom this, it follo w s that the n u m b ers giv en by (1) corr esp ond to sequ en ce A 00 0111 in Sloane [11] and are kno wn as E uler numb ers . 2 trees # cherries # pitc hforks 4 3 1 2 5 1 2 5 4 3 1 5 3 4 2 3 1 2 4 5 3 0 5 4 4 5 3 2 1 3 2 1 4 5 1 2 3 1 3 4 5 2 2 1 2 3 4 1 5 3 4 2 5 1 3 2 4 5 1 2 1 5 4 1 2 3 5 1 2 4 3 1 2 3 5 4 2 2 1 2 5 3 4 2 0 2 3 4 5 1 1 1 Figure 1: The sixteen p ossible ranked trees o f size s ix c la ssified by shap e. Within each class all possible ordering s of the internal no des are display ed. Num b er of cherries a nd pitchforks are indicated. 3 2.1 T rees as a result of t he coalescen t pro cess The coalesce nt of size n is a mo del for the genealogic al history of a sample of n genes. It has b een introdu ced in p opulatio n genetics by Kingman an d Ew en s [7, 8] and has no wada ys textbo ok status [13]. Rank ed trees can b e generated b y the c o alesc ent pr o c ess , which starts with n lea ves and w orks b y successiv ely coalescing tw o randomly c h osen b ranc h es until it reac hes the ’most recen t common ancestor’ when the last t wo remaining b ranc hes are joined. T o reflect time order one can assign an integ er to eac h internal no de when created, for instance the lab el n − 1 to the firs t coalesc en t ev ent and 1 to the last even t, th e most recent common ancestor, or the ro ot of the tree. The p robabilit y distribution of r an ked trees P R generated u n der the coalesce nt pro cess is essent ially conta in ed in the p ap er of T a jima [12] and it is describ ed b el o w. Probabilit y distribution of ra nk ed trees Let t ∈ R and let o ( t ) b e the n umb er of internal no des i whose children are t w o lea v es. Su c h inte rnal n o des are called the cherries of the tree. F o r example, (see Fig. 1). Giv en t ∈ R n , f rom T a jima [12] follo ws that P R ( t ) = 2 n − 1 − o ( t ) ( n − 1)! , (2) i.e. the probab ility of any r an ked tree t ∈ R n dep ends only on t wo parame- ters, o and n . The probability of generating the same rank ed trees twice Considering trees link ed on a common c hromosome one observes that c hromosomal link age substantia lly increases the probabilit y that t wo ’neigh- b oring’ trees are id en tical ev en if separated by a recombinatio n ev en t. T o quan tify the effect of link age and recom bin ation it is imp ortan t to kno w the bac kground probabilit y that t wo indep endently generated tr ees are identica l. This probabilit y can b e f ound with the help of the genarating fun ction Y ( x, z ) = X t ∈R x o ( t ) z n ( t ) − 1 ( n ( t ) − 1)! , discussed in more details in Section 3.1.1, eq. (6). W e h a v e the follo win g result. 4 Prop osition 1 Th e pr ob ability that two indep endently gener ate d r anke d tr e es of size n ar e identic al is p n = 4 n − 1 ( n − 1)! × [ z n − 1 ] Y  1 4 , z  . Pr o of: F rom eq. (2) the probability that t 1 , t 2 ∈ R n are identi cal is p n = X t ∈R n P R ( t ) 2 = 1 ( n − 1)! 2 X t ∈R n 4 n − 1 − o ( t ) = 4 n − 1 ( n − 1)! 2 X t ∈R n  1 4  o ( t ) = 4 n − 1 ( n − 1)! × [ z n − 1 ] Y  1 4 , z  , where [ z n − 1 ] Y (1 / 4 , z ) means the ( n − 1)-st coefficient of the T a ylor ex- pansion of Y (1 / 4 , z ) in z = 0.  3 En umerativ e results 3.1 Outdegree of the no des i n rank ed and 0 - 1 - 2 -increasing trees Let t ∈ R n and m = n − 1. Remo ve all leav es and external branc h es f rom t and obtain a reduced tree ρ ( t ). Th e tree ρ ( t ) is a so-called 0 - 1 - 2 -incr e asing tr e e of size m , where, this time, the size is the total num b e r of no des in the tree and not only of the lea ves. Th e class I 012 of 0-1-2-i ncreasing trees is comp osed of un-ordered ro oted trees where all no des ha v e outdegree 0, 1 or 2. The m no des of suc h a tree carry totall y ordered lab els b elonging to { 1 , 2 , ..., m } . Moreo v er, the lab elling is suc h that any c h ild no de lab el is greater than that of the parent no d e. As usual I 012 m denotes the set of 0-1-2- increasing trees of size m . Hence, the function ρ is a bijection from R n to I 012 m . Giv en a r ank ed tree t , the outde gr e e of an internal no de of t is the outdegree of the corresp ond ing no de in ρ ( t ). Th u s, if t ∈ R , the no d es of outdegree 0 (resp. 1, 2) are defined as the no des with 2 (resp. 1, 0) lea v es as d irect descend ants. 5 1 1 2 1 2 3 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 1 2 3 4 Figure 2: Firs t levels o f the generating tree asso ciated to Θ. Here, we deriv e th e enumeration of 0-1-2 -increasing tr ees with resp ect to the size and to the n umb er of no d es with outdegree 0,1 and 2. The bijection ρ will allo w us to use this en umerativ e r esult in S ection 3.1.2 to determine the pr obabilit y distrib ution of the random v ariable o , the numb er of cherries , when t is a ranke d tree of s ize n generated by the coalescen t pro cess. It is already kno wn (see McKenzie [ 9]) that o ( t ) is asymptotically normal for large n . 3.1.1 Recursiv e construction of 0 - 1 - 2 -increasing trees W e sh o w no w ho w the class of 0-1-2- increasing trees can b e generated r ecur- siv ely . In p articular w e construct eac h tree b elo nging to I 012 m +1 b y adding a new no de to some tree in I 012 m . Th is construction, denoted by Θ, will then b e translated into a fun ctional equation. Solving the equation w e obtain a biv ariate exp onentia l generating function counting the considered increasing trees with resp ect to size and to the n u m b er of nod es with outdegree 0, 1 and 2. Giv en a tree t ∈ I 012 m , Θ simply adds the no d e lab elled ’ m + 1’ as a c h ild of a no de of t having outdegree less than t wo . Let o ( t ) , p ( t ) and q ( t ) denote the n umb er of no des with outdegree 0 , 1 and 2 resp ec tiv ely . Θ applied to t pr o duces o ( t ) + p ( t ) elemen ts of I 012 m +1 eac h time add ing the new no d e lab elled m + 1 as a c h ild of the no des coun ted in o ( t ) + p ( t ). In Fig. 2 we depict the fir st steps of this construction p r o cess. Note that o ( t ) = q ( t ) + 1 and o ( t ) + p ( t ) + q ( t ) = m . F rom th ese relations w e h a ve , in particular, that p ( t ) = m − 2 o ( t ) + 1. The construction Θ can b e translated into the follo wing suc c ession rule (see Band erier et al. [1]) where eac h tree is repr esented b y a lab el comp osed of th e v alues of its p arameters 6 o and m while the exp onents sho w ho w man y times the lab el is pr o duced, ( o, m ) → ( o, m + 1) o ( o + 1 , m + 1) m − 2 o +1 . In particular, giv en a tree t with parameters o = o ( t ) and m = m ( t ), the application of Θ to t pr o duces o new trees ha ving size m + 1 and o c h erries and m − 2 o + 1 new trees h a ving size m + 1 and o + 1. T h e s tarting p oint of th e construction is the unique tree of size one represented by (1 , 1). No w consider the exp onen tial generating function Y ( x, z ) = X t ∈I 012 x o ( t ) z m ( t ) m ( t ) ! . The p r evious succession rule can b e translated as follo w s in to an equ ation for Y ( x, z ). Y ( x, z ) = xz + X x o z m ∈I 012 ox o z m +1 ( m + 1) ! + X x o z m ∈I 012 ( m − 2 o + 1)( x o +1 z m +1 ) ( m + 1) ! = xz + (1 − 2 x ) X x o z m ∈I 012 ox o z m +1 ( m + 1) ! + xz X x o z m ∈I 012 x o z m m ! = xz + (1 − 2 x ) X x o z m ∈I 012 ox o z m +1 ( m + 1) ! + xz Y ( x, z ) F rom the previous equation we obtain that Y ( x, z )(1 − xz ) − xz 1 − 2 x = X x o z m ∈I 012 ox o z m +1 ( m + 1) ! . Differen tiating b oth sides with resp ect to the v ariable z we hav e 1 1 − 2 x  d Y dz ( x, z ) (1 − xz ) − x Y ( x, z ) − x  = x d Y dx ( x, z ) , whic h is equiv alen t to x (1 − 2 x ) d Y dx ( x, z ) + ( xz − 1) d Y dz ( x, z ) = − x Y ( x, z ) − x. (3) 7 The previous first order partial differen tial equation can b e solv ed using the metho d of char acteristics (see [2 ]) resp ecting th e condition give n by eq. (1) Y (1 , z ) = sec( z ) + tan( z ) − 1 . Indeed Y (1 , z ) m u s t repr esent the exp onentia l generating function counting 0-1-2- increasing trees with resp ect to size. Applying the metho d consists, fir s t, of solving the t wo follo w ing ordinary differen tial equations z ′ = xz − 1 x (1 − 2 x ) Y ′ = − xY − x x (1 − 2 x ) The solutions are z = c 1 + 2 arctan( √ 2 x − 1) √ 2 x − 1 , Y = c 2 √ 2 x − 1 − 1 , (4) with constants c 1 and c 2 and wh er e c 2 can b e written as a function of c 1 in the follo wing wa y c 2 = G ( c 1 ) = G ( z √ 2 x − 1 − 2 arctan( √ 2 x − 1)) . In th is wa y equation (4 ) b ecomes Y ( x, z ) = G ( z √ 2 x − 1 − 2 arctan( √ 2 x − 1)) √ 2 x − 1 − 1 , whic h give s sec( z ) + tan( z ) − 1 = Y (1 , z ) = G ( z − π 2 ) − 1 . F unction G m ust satisfy G ( z ) = sec( z + π 2 ) + tan( z + π 2 ) = − 1 − cos( z ) sin( z ) . Inserting this into (4) w e ha ve 8 Y ( x, z ) = √ 2 x − 1  − 1 − cos( z √ 2 x − 1 − 2 arctan( √ 2 x − 1)) sin( z √ 2 x − 1 − 2 arctan( √ 2 x − 1))  − 1 , whic h , after some calculations, fin ally giv es Y ( x, z ) = √ 2 x − 1 tan  − z √ 2 x − 1 2 + arctan( √ 2 x − 1)  − 1 . (5) Note th at the condition Y (1 , z ) = sec( z ) + tan( z ) − 1 is resp ected. Indeed Y (1 , z ) = 1 tan  − z 2 + π 4  − 1 and 1 tan  − z 2 + π 4  = 1 + tan  z 2  1 − tan  z 2  = 1 + cos( z ) + sin( z ) 1 + cos( z ) − sin( z ) = 1 + cos 2 ( z ) + 2 cos( z ) − sin 2 ( z ) (1 + cos( z ) − sin( z )) 2 = cos( z ) 1 − sin( z ) = 1 + sin ( z ) cos( z ) Moreo v er, using the f act that exp ( z √ − 2 x + 1) = cos( z √ 2 x − 1) + i sin( z √ 2 x − 1) , w e can write eq. (5) in terms of th e exp onenti al function as Y ( x, z ) = 2  x exp  √ − 2 x + 1 z  − x   √ − 2 x + 1 − 1  exp  √ − 2 x + 1 z  + √ − 2 x + 1 + 1 . (6) P erform ing th e substitution x = 1 / 4 we ha ve th at Y  1 4 , z  = e  q 1 2 z  − 1 2   q 1 2 − 1  e  q 1 2 z  + q 1 2 + 1  , 9 the T a ylor expansion of whic h is Y  1 4 , z  = 1 4 z + 1 8 z 2 + 5 96 z 3 + 1 48 z 4 + 1 120 z 5 + . . . . Using th e result of Pr op osition 1 we can no w effectiv ely calculate the proba- bilit y p n that t w o ranked trees having n lea v es are iden tical when generated indep end en tly b y the coalesce nt pro cess: p 2 = 4 1! × 1 4 = 1 , p 3 = 4 2 2! × 1 8 = 1 , p 4 = 4 3 3! × 5 96 = 5 9 , p 5 = 4 4 4! × 1 48 = 2 9 and p 6 = 4 5 5! × 1 120 = 16 225 , an d so on. 3.1.2 The probability dis tribution of the n u m b e r of c herries W e are n o w ready to state th e enumeration of ranked trees with resp ect to size and n u m b er of no des of outdegree 0, 1 or 2, wh en eac h tree is weigh ted by its pr ob ab ility under the coalescen t p ro cess. This exact en u merativ e result is no ve l and ac hieved with the help of the w eigh ted generating function F ( x, z ) = X t ∈R n , n> 1 2 n ( t ) − 1 − o ( t ) ( n ( t ) − 1)! x o ( t ) z n ( t ) . F unction F has a more in tuitiv e in terp retation if one considers the trans- formation Y w = F z instead. It can b e in terpreted as a w eighte d exp onential generating function counting 0-1-2-increa sing trees with resp ect to the out- degree and the total num b er of no des. Starting from equation (6), we p erform some sub stitutions on Y to ob tain Y w . In particular w e h a v e Y w = Y  x 2 , 2 z  and, multiplying by z , we fi nally obtain th e desir ed function F . Prop osition 2 Th e weighte d or dinary gener ating function of r anke d tr e es c onsider e d with r esp e ct to size and numb er of cherries is F ( x, z ) = z x exp  2 z √ − x + 1  − z x  √ − x + 1 − 1  exp  2 z √ − x + 1  + 1 + √ − x + 1 . (7) The pr ob ability of having o ′ cherries in a r anke d tr e e of size n c orr esp onds to the c o effic i ent of x o ′ z n in the T aylor exp ansion of F ar ound z = 0 , i.e. P n ( o = o ′ ) = [ x o ′ z n ] F ( x, z ) . The fi rst terms of th e T a ylor expans ion of (7) are describ ed b elo w ; 10 F ( x, z ) = xz 2 + xz 3 + 1 3  x 2 + 2 x  z 4 + 1 3  2 x 2 + x  z 5 + 1 15  2 x 3 + 11 x 2 + 2 x  z 6 + 1 45  17 x 3 + 26 x 2 + 2 x  z 7 + 1 315  17 x 4 + 180 x 3 + 114 x 2 + 4 x  z 8 + . . . . Lo oking at Fig. 1 one can c h ec k th at, for example, there are exactly 11 trees repr esented by th e monomial x 2 z 6 . Eac h one of them h as probabilit y 1 15 . This is in agreement with the term 11 15 x 2 z 6 in th e exp an s ion. Indeed, 11 15 is the probabilit y to obtain a r ank ed tree of size 6 with t wo cherries. Using the result of Pr op osition 2 w e compute the discrete probability distribution of the random v ariable o ( t ) for trees of fixed size n . In th is case o is a r an d om v ariable whic h tak es v alues b et we en 1 and ⌊ n/ 2 ⌋ . In Fig. 3 w e hav e depicted the distribution of o for a rank ed tree of size n = 54. By Prop osit ion 2 one can also d etermine the exp ec ted v alue E o ( n ) and the v ariance V ar o ( n ) of the ran d om v ariable o in dep en d ence of tree size n . Using other metho ds these ha ve b ee n d etermined b efore, for example b y McKenzie [9]. Using our appr oac h the exp ectation is E o ( n ) = [ z n ] dF dx (1 , z ) = [ z n ] z 4 − 3 z 3 + 3 z 2 3( z − 1) 2 . If n > 2, this simplifies to E o ( n ) = n 3 . 11 The second moment is E o 2 ( n ) = [ z n ] d ( x dF dx ) dx (1 , z ) = [ z n ] d 2 F dx (1 , z ) + [ z n ] dF dx (1 , z ) = [ z n ] 2( z 7 − 6 z 6 + 15 z 5 − 15 z 4 ) 45( z − 1) 3 + E o ( n ) = [ z n ]  2 ( z − 1) 3  z 7 45 − 2 z 6 15 + z 5 3 − z 4 3  + E o ( n ) . If n > 6, and using V ar o ( n ) = E o 2 ( n ) − E 2 o ( n ), we obtain th e v ariance of o V ar o ( n ) = − ( n − 5)( n − 6) 45 + 2( n − 4)( n − 5) 15 − ( n − 3)( n − 4) 3 + ( n − 2)( n − 3) 3 + n 3 − n 2 9 = 2 n 45 . Note that this is the v ariance of c herries of indep endently generated trees. Considerin g ’linke d’ trees, i.e. along a recom binin g c hromosome, th e v ariance is smaller. 3.2 The n um b er of pitc hforks The recursive construction presen ted in Section 3.1.1 ca n b e extended in order to consider also pitchforks . Using differen t m etho d s, they hav e b een studied b efo re for example by Rosen b erg [10]. A p itc hfork in a rank ed (resp. 0-1-2-increasing) tree is simply a subtr ee with 3 lea v es (resp . 2 no des). If r ( t ) d en otes the num b er of pitc hforks in t ∈ I 012 the construction of Section 3.1.1 is extended to the new r andom v ariable r . W e fin d the f ollo wing succession rule: ( o, r, m ) → ( o, r , m + 1) r ( o, r + 1 , m + 1) o − r ( o, r, m ) → ( o + 1 , r − 1 , m + 1) r ( o + 1 , r , m + 1) m − 2 o +1 − r . Considering no w Y ( x, v , z ) = X t ∈I 012 x o ( t ) v r ( t ) z m ( t ) m ( t ) ! , 12 w e obtain the follo wing different ial equation: ( v + x )( v − 1) d Y dv = x + xY + x ( v − 2 x ) d Y dx + ( xz − 1) d Y dz . (8) F or v = 1 it reduces to eq. (3) bu t there is non easy analytic solution. Ho we v er, we can still obtain the exp ected v alue E r ( m ) for the num b er of pitc hf orks in 0-1-2 increasing trees with m no des. S tarting f r om (8 ) and p erforming the sub stitutions x = 1 / 2 and z = 2 z we obtain d Y dv  1 2 , v , 2 z  = 1 + Y  1 2 , v , 2 z  + 2( z − 1) d Y dz  1 2 , v , 2 z  2  v + 1 2  ( v − 1) + d Y dx  1 2 , v , 2 z  2  v + 1 2  from w h ic h w e ha v e [ z m ] d Y dv  1 2 , v , 2 z  = [ z m ] Y  1 2 , v , 2 z  + 2( z − 1) d Y dz  1 2 , v , 2 z  2  v + 1 2  ( v − 1) +[ z m ] d Y dx  1 2 , v , 2 z  2  v + 1 2  . When v → 1 we find that E r ( m ) = [ z m ] lim v → 1 Y  1 2 , v , 2 z  + 2( z − 1) d Y dz  1 2 , v , 2 z  2  v + 1 2  ( v − 1) ! + 2 E o ( m ) 3 . The considered limit can b e d etermined acc ording to l’ H ospital’s rule taking the deriv ativ e of the numerator and the denominator with resp ect to v and p erf orm ing then th e substitution v = 1. F urthermore, from Section 3.1.2 E o ( m ) = ( m + 1) / 3, and thus 13 E r ( m ) = [ z m ]   1 3 X t ∈I 012 r ( t ) 2 m ( t ) − o ( t ) m ( t ) ! z m ( t )   +[ z m ]   2 3 ( z − 1) X t ∈I 012 r ( t ) m ( t ) 2 m ( t ) − 1 − o ( t ) m ( t ) ! z m ( t ) − 1   + 2( m + 1) 9 = 1 3 E r ( m ) + [ z m ] z − 1 3 z X k > 0 k E r ( k ) z k ! + 2( m + 1) 9 = 1 3 E r ( m ) + mE r ( m ) − ( m + 1) E r ( m + 1) 3 + 2( m + 1) 9 . Reordering terms we obtain the recursion E r (2) = 1; ( m + 1) E r ( m + 1) = ( m − 2) E r ( m ) + 2( m + 1) 3 . This giv es for an increasing tree with m > 2 n o des E r ( m ) = m + 1 6 . F rom eq. (8) one ca n also compute the full p robabilit y distribution of the random v ariable r when an increasing tree of fix ed size is generated by the coal escen t pro cess. Indeed, if we consider Y m ( x, v , z ) = X t ∈I 012 m x o ( t ) v r ( t ) z m m ! the follo w in g r esult pro vides a recur sion whic h can b e used to compu te the functions Y m for an y m ≥ 1. Prop osition 3 Th e fol lowing r e cursion holds: Y 1 = xz Y m +1 = Z  ( v + x )(1 − v ) d Y m dv + xY m + x ( v − 2 x ) d Y m dx + xz d Y m dz  dz 14 Pr o of. Consider eq. (8 ) w ithout the monomial x whic h app ears there. If we then isolate the term d Y dz and in tegrate b oth sides of the resulting equation with resp ect to th e v ariable z w e obtain the p olynomia l Y m +1 starting from Y = Y m .  The r esults for m = 1 , 2 , 3 , 4 , 5 are as follo ws Y 1 = xz Y 2 = 1 2 v xz 2 Y 3 = 1 6 v xz 3 + x 2 z 3 6 Y 4 = 1 24 v xz 4 + x 2 z 4 24 + 1 8 v x 2 z 4 Y 5 = 1 120 v xz 5 + x 2 z 5 120 + 7 120 v x 2 z 5 + 1 40 v 2 x 2 z 5 + x 3 z 5 30 The ab ov e r esults concerning cherries and pitc hforks can b e extended to the join t and conditional pr obabilit y distribu tions (see Fig. 4). Summarizing, w e state Prop osition 4 i) The pr ob ability of having r ′ pitchforks in an incr e asing tr e e of size m (se e Fig. 3) i s P m ( r = r ′ ) = [ v r ′ ] Y m  1 2 , v , 2  ; ii) The pr ob ability of having o ′ cherries and r ′ pitchforks in an inc r e asing tr e e of size m is P m ( o = o ′ , r = r ′ ) = [ x o ′ v r ′ ] Y m  x 2 , v , 2  ; iii) The pr ob ability of having r ′ pitchforks in an incr e asing tr e e of size m given it has o ′ cherries (se e Fig. 4) is P m ( r = r ′ | o = o ′ ) = P m ( o = o ′ , r = r ′ ) P m ( o = o ′ ) = [ x o ′ v r ′ ] Y m  x 2 , v , 2  [ x o ′ ] Y m  x 2 , 1 , 2  . 15 0 5 10 15 20 25 number 0 0.05 0.1 0.15 0.2 0.25 probability pitchforks cherries Figure 3: Distributions of cherries and pitc hfo r ks for R 54 (i.e. I 012 53 ). 0 5 10 15 20 25 number cherries 0 5 10 mean (+-stdev) pitchforks Figure 4 : Mean of the conditional probability distribution of pitchforks given the nu m ber o f cherries for R 54 . 16 Ac knowledgme n ts W e gratefully ac kno wledge helpful discus s ions with L. F erretti, A. Klass- mann and A. Malina. Financial supp ort wa s pro vided by the German Re- searc h F oundation (DF G-SFB680) . References [1] C. Banderier, M. Bousquet-Melo u, A. Denise, P . Fla jolet, D. Gardy , and D. Gouy ou-Beauc hamps. Generating functions for generating trees. In Pr o c e e dings of 11-th formal p ower series and algebr aic c ombinato rics , pages 40–52, 1999. [2] R. Courant, D. Hilb ert. Metho ds of Mathematic al Physics . John Wiley & S on s , Inc., 1989. [3] P . Fla jolet and R. Sedgewic k . Ana lytic Combina- torics . Cam b r idge Unive r sit y Press, 2009. URL http://a lgo.inria .fr/flajolet/Publications/books.html . [4] E. F. Harding. The p robabilities of r o oted tree-shap es generated b y random bifur cation. A dvanc es in Applie d Pr ob ability , 3(1):pp. 44–77, 1971. IS S N 00018 678. URL http://w ww.jstor. org/stable/1426329 . [5] Hud son, R. R. (1990). Gene genealogies and the coalescen t pro cess. In Oxf ord Su rv eys in Evo lutionary Biolo gy vol. 7, pp. 1–44. Oxford Univ ersity Pr ess. [6] R. R. Hudson. Generating samples under a Wright -Fisher neutral mo del of genetic v ariation. Bioinformatics , 18:337 –338, 2002. [7] J. F. C . K ingman. The coalescen t. Sto chastic Pr o c e sses and their Ap- plic ations , 13:23 5–248 , 1982 . [8] J. F. C. Kingman. Origins of the coalescen t. 1974-1 982. Genetics , 156 (4):14 61–14 63, Dec 2000. [9] A. McKenzie and M. Steel. Distrib utions of cherries for t wo m o dels of trees. Mathematic al B i oscienc e s , 164:81–92 , 2000. [10] N.A. Rosen b erg. The mean and the v ariance of the num b ers of r- pronged n o des and r-cat erpillars in Y ule generated genealo gical tr ees. Anna ls of Combinatorics , 10:129–14 6, 200 6. 17 [11] N. J. A. Sloane. The on-line encyclop edia of in teger sequen ces. Notic es Amer. Math. So c. , 50(8): 912–9 15, 2003. ISSN 0002-9920. [12] F. T a jima. Evo lutionary relationship of DNA sequences in finite p op- ulations. Genetics , 105(2 ):437–460, Oct 1983 . [13] J . W ak eley . Co alesc ent the ory – an intr o duction . Rob erts&Company , Green woo d Village, Colorado, 2009. 18