On edge-weighted recursive trees and inversions in random permutations.

Transcription

1 O edge-weighted recursive trees ad iversios i radom permutatios. M. Kuba ad A. Paholzer Istitut für Diskrete Mathematik ud Geometrie Techische Uiversität Wie Wieder Hauptstr. 8-0/ Wie, Austria, Markus.Kuba@tuwie.ac.at, Alois.Paholzer@tuwie.ac.at Correspodig Author: M. Kuba May 30, 2006 Abstract We itroduce radom recursive trees, where determiistically weights are attached to the edges accordig to the labelig of the trees. We will give a biectio betwee recursive trees ad permutatios, which relates the arisig edge-weights i recursive trees with iversios of the correspodig permutatios. Usig this biectio we obtai exact ad limitig distributio results for the umber of permutatio of size, where exactly m elemets have iversios. Furthermore we aalyze the distributio of the sum of labels of the elemets, which have exactly iversios, where we ca idetify Dickma s ifiitely distributio as the limit law. Moreover we give a distributioal aalysis of weighted depths ad weighted distaces i edgeweighted recursive trees. Keywords: Recursive tree, permutatio, iversio, Dickma distributio 2000 Mathematics Subect Classificatio 05C05. Itroductio There are several well-kow biectios betwee recursive trees ad permutatios. For example, a rotatio correspodece is give i [2], which immediately characterizes the distributio of the root degree, the umber of leaves, etc., i a radom recursive tree. Here we state a atural, but ew biectio (best to our kowledge), which maps iversios i radom permutatios of {, 2,..., } to suitably defied weights o the edges of recursive trees with odes. Usig this biectio we are able to relate parameters i permutatios, as, e.g., the umber of iversios, with parameters i recursive trees. A rooted o-plae size- tree labeled with distict itegers, 2,..., is a recursive tree if the ode labeled is distiguished as the root, ad, for each 2 k, the labels of the odes o the uique path from the root to the ode labeled k form a icreasig sequece. Every size recursive tree ca be obtaied uiquely by attachig ode to oe of the odes i a recursive tree of size. This immediately shows that the umber T of recursive trees of size is give by ( )!, for. Throughout this paper we assume as the model of radomess the radom tree model, which meas that all ( )! recursive trees of size are cosidered to appear equally likely. We speak the about radom recursive trees. Equivaletly oe may describe radom recursive trees via the followig tree evolutio process, which geerates radom recursive This work was supported by the Austria Sciece Foudatio FWF, grat S9608.

2 trees of arbitrary size. At step the process starts with the root labeled by. At step i + the ode with label i + is attached to ay previous ode v of the already grow tree T of size i with probability p i (v) /i. Due to this simple growth rule radom recursive trees have bee itroduced as a probability model i several areas. E.g., they are used to model the spread of epidemics [6], to aid i the costructio of the family trees of preserved copies of aciet mauscripts [7], or to model chai letter ad pyramid schemes [6]. Furthermore they are used to model the stochastic growth of etworks [3]. See also the survey paper [5]. Let T deote a recursive tree of size T. Throughout this paper we always cosider edgeweighted recursive trees, where every edge e E E(T ) of the tree will be weighted determiistically as follows. If the edge e (, k) is adacet to the odes ad k, the we defie the weight w e of the edge e as w e : k. The aim of this paper is to study several parameters of radom recursive trees appearig i coectio with this edge-weights. We will aalyze the radom variable S : e E w e coutig the sum of all edge-weights ad the radom variable S, e E {we} coutig the umber of edge-weights w e with w e i a radom recursive tree of size. By usig the biectio betwee recursive trees ad permutatios already metioed it turs out that the r. v. S essetially couts the umber iversios ad the r. v. S, is give by the umber of elemets with exactly iversios i a radom permutatio of size. We also study the r. v. W, : e(e,e 2) E e 2 {we}, which couts the sum of the labels of odes attached via edges of weight, for the full rage of () growig with the size of a radom recursive tree. We show that the limitig distributio of W, depeds o the growth of (), where Dickma s ifiitely divisible distributio appears i the limit law. Furthermore we aalyze the radom variable G coutig the umber of abset edge-weights of the set {, 2,..., } i a radom recursive tree of size. We ca show a Gaussia limit law for the suitably ormalized ad cetered r. v. G. Moreover we cosider the radom variable M, which gives the maximum edge-weight appearig i a radom recursive tree of size. Due to the correspodece with iversio statistics i radom permutatios we obtai a Rayleigh distributio as limit law of the suitably ormalized ad scaled r. v. M. I a rooted tree the depth of ode v, also called the level of ode v, is measured by the umber of edges lyig o the uique path from the root to ode v. Here we cosider a geeralizatio of the depth for edge-weighted trees. Let X, : e E w e (A e ) deote the radom variable coutig the edge-weighted depth of ode i a radom recursive tree of size, where A e deotes the evet that edge e is o the uique path from the root to ode. Further we deote by X, the radom variable coutig the edge-weighted distace betwee odes ad i a radom recursive tree of size, measured by the sum of the edge-weights o the uique path from to. The r. v. X, ca be trivially characterized, whereas the characterizatio of X, leads to a discrete limit law. Usig the distributio law of X, we are also able to obtai the distributio of the label of the root of the spaig tree of two radomly selected odes i a radom recursive tree of size. Whe speakig about iversios i permutatios we will here always thik about right iversios, i.e., iversios caused by elemets to the right. For a permutatio σ (σ... σ ) of {, 2..., } ad k we call the umber of elemets i σ to the right of k, which are smaller tha k, the elemet iversios i k i k (σ) of k. Hece the iversio table of σ (σ... σ ) is give by (i, i 2,..., i ), with the restrictios 0 i k k, for k. Throughout this paper we deote by X Y the equality i distributio of the radom variables X ad Y, ad with X X the weak covergece, i. e., the covergece i distributio, of the sequece of radom variables X to a radom variable X. For idepedet radom variables X ad Y we deote the sum of X ad Y by X Y, whereas for ot ecessarily idepedet radom variables X ad Y we write X + Y. We will deote by [ k] the sigless Stirlig umbers of the first kid ad by k the Euleria umbers. Furthermore we use the Iverso bracket-otatio: for a statemet A we have [[A]] if A is true ad [[A]] 0 otherwise. Moreover we deote by { < c k} the evet that ode is attached to ode k (i.e., is a child of k) for a give tree T. 2

3 I Sectio 2 we give the biectio betwee recursive trees ad iversios i permutatios ad study the radom variables S, G ad M. Sectio 3 is devoted to the aalysis of specific edge weights S, ad W,, whereas Sectio 4 is devoted to the aalysis of edge-weighted depths ad distaces, X, ad X,. 2 Edge-weights ad iversios i radom permutatios 2. A biectio betwee recursive trees ad permutatios We preset the followig biectio betwee recursive trees of size ad permutatios of {,..., }, which turs out to be appropriate whe studyig parameters i edge-weighted recursive trees. Biectio. Cosider a recursive tree T of size ad its edge set E. We eumerate the edges of E by e 2, e 3,..., e, where e k, with k 2, is defied as the edge e k (, k) coectig ad k, with k. The edge e k is uiquely defied, sice every ode k 2 i a recursive tree is attached to exactly oe ode with k. We defie ow the umbers q k : w ek k as the edge-weight of edge e k ad cosider the edge-weight table (q 2, q 3,..., q ) of T. Of course, it holds q k k, for 2 k. If we defie umbers i k : q k+, for k, the it holds 0 i k k ad the array (i, i 2,..., i ) correspods to the iversio table of a permutatio σ of {, 2,..., }, which uiquely determies σ. To costruct a recursive tree T of size from a give permutatio σ of {,..., } with iversio table (i,..., i ) oe starts with as the root of T ad attaches successively ode k, with 2 k, to ode k i k, which leads to a edge-weight table (q 2,..., q ) with q k k i k +, for 2 k. 2.2 The sum of edge-weights Theorem. The distributio of the radom variable S, coutig the sum of the edge-weights i a radom recursive tree of size, is give by S ξ ( ) U U 2 U, () where the radom variable ξ couts the umber of iversios of a radom permutatio of size, ad U k deotes a uiform distributio o the set {, 2,..., k}. Proof. The theorem is a immediate cosequece of Biectio, sice for a give recursive tree of size with edge-weight table (q 2,..., q ) ad iversio table (i,..., i ) of the correspodig permutatio of size we always have 2 k q k + k i k. Furthermore for a radom recursive tree of size the edge-weights q k are uiformly distributed o {,..., k } idepedet of the edge-weights q l, l k, sice k is attached to oe of the odes k at radom. Usig Theorem ad the cetral limit theorem for the umber of iversios ξ of a radom permutatio of size, see [9], we obtai the followig corollary. Corollary. The properly scaled ad shifted radom variable S coverges, for, i distributio to a stadard ormal distributed r. v.: S : S E(S ) V(S ) where N (0, ) deotes the stadard ormal distributio. N (0, ), (2) 3

4 2.3 The maximal edge-weight Let M deote the radom variable, which gives the maximal edge-weight i a radom recursive tree of size. As a immediate cosequece of Biectio we obtai the followig result. Theorem 2. The distributio of the r. v. M + is for give as follows. M + η max{u, U 2,... U }, (3) where the radom variable η gives the maximal etry i the iversio table of a radom permutatio of size, ad U k deotes a uiform distributio o the set {, 2,..., k} ad all r. v. U k are mutually idepedet. We wat to remark that, due to Theorem 2, M + also couts the umber of passes that are required to sort a radom permutatio of size by the sortig algorithm bubble sort. Usig results from [9] ad [4] we obtai distributioal results for M +. Corollary 2. The exact distributio of M is give as follows. P{M + m} m! m m, for m, ad.! The limitig distributio of the suitably scaled ad shifted r. v. M ca be characterized as follows: M : M X, where X is a Rayleigh distributed radom variable with desity fuctio f(x) xe x2 2, for x 0, ad f(x) 0, otherwise. 2.4 The umber of abset edge-weights Let G deote the radom variable coutig the umber of elemets of the set {, 2,..., }, which do ot appear as a edge-weight i a radom recursive tree of size. Usig Biectio we immediately get the followig result. Theorem 3. G is distributed as the umber of elemets of the set {0,,..., 2}, which are ot appearig i the iversio table of a radom permutatio of size. Furthermore the followig distributioal equatio holds: G {, 2,..., } \ {U, U 2,..., U }, (4) where U k deotes a uiform distributio o {, 2,..., k} ad all radom variables are mutually idepedet. We will study ow the distributio of G, where it will tur out that it is slightly more coveiet whe cosiderig G +. Obviously we have P{G + 0} P{G + } /!, for, which correspods to a star-like tree: {k < c } for k, or a chai: {k + < c k} for k. Furthermore it holds that P{G + } 0, for. Whe distiguishig whether the weight of the edge adacet to ode + is occurrig also amogst the edge-weights of the remaiig edges or ot, we ca set up easily the followig recurrece for G + : P{G + m} m P{G m } + m + P{G m}, for m ad 2. (5) We itroduce ow, for 0 m ad, the umbers a,m : P{G + m}t + P{G + m}!. We obtai the the margi values a,0 ad a, 0, for, ad for m ad 2 the recurrece a,m ( m)a,m + (m + )a,m. (6) 4

5 But this is exactly the recurrece for the Euleria umbers m coutig the umber of permutatios of size with exactly m ascets: m ( m) + (m + ) m m. (7) Note that a ascet i the permutatio σ (σ, σ 2,..., σ ) is defied as a positio, with, such that σ < σ +. This immediately leads to the followig result. Theorem 4. The distributio of the radom variable G + is for give as follows: m P{G + m} where m deote the Euleria umbers.!, for 0 m, (8) Usig Theorem 4 ad the cetral limit theorem for the Euleria umbers [] we obtai the limit distributio of G. Corollary 3. The properly scaled ad shifted radom variable G coverges, for, i distributio to a stadard ormal distributed r. v.: G : G E(G ) V(G ) where N (0, ) deotes the stadard ormal distributio. N (0, ), (9) Of course the previous cosideratios lead to the followig corollary, which seems to be ew best to our kowledge. Corollary 4. The umber of permutatios of {, 2,..., } with exactly m ascets ad the umber of permutatios of {, 2,..., } with exactly m abset values i its iversio table (i.e., exactly m elemets of the set {0,,..., }, which are ot appearig i its iversio table) coicide ad are give by the Euleria umbers m. Remark. We wat to poit out that oly the umber of permutatios coicide i Corollary 4. It is ot true i geeral that a permutatio with exactly m ascets also has m abset values i its iversio table. Cosider, e.g., the permutatio σ (2, 4,, 3) with 2 ascets, but whe cosiderig its iversio table (0,, 0, 2) we see that oly oe value, amely 3, is abset. 3 The distributio of specific edge-weights 3. The umber of edge-weights of a give size Let us deote by E the set of edges of a edge-weighted radom recursive tree of size. We will study the radom variable S, : e E {we}, which couts the umber of edges with weight i a size- radom recursive tree. This ca be doe by showig relatios to the ode degree of specified odes i radom recursive trees. At first we give the followig immediate cosequece of Biectio. Propositio. S, I,, for >, (0) where I, deotes the r. v., which couts the umber of elemets i a radom permutatio of size with exactly iversios. The ext theorem gives a relatio betwee the distributio of edges with a give weight ad the out-degree of specified odes i recursive trees ad characterizes the distributio appearig. 5

6 Theorem 5. The radom variable S, satisfies the followig distributio law. S, Z, B B, () where the radom variable Z, gives the out-degree (i.e., the umber of childre, or the the umber of attached odes) of ode i a radom recursive tree of size, ad B k deotes a Beroullidistributed r. v. Be( k ), i.e., P{B k } /k ad P{B k 0} /k. Proof. We obtai the followig distributioal equatio of S, : S, k+ {k<ck }, (2) where {k < c i} deotes the evet that ode k is attached to ode i (i.e., k is a child of i). These idicators are mutually idepedet for recursive trees ad by defiitio give via P{k < c i} /(k ), for i k. Thus S, k Be( k ). To establish the coectio to the out-degree Z, of ode i a size- radom recursive tree we use the followig distributioal equatio: Z, k+ Sice P{k < c } /(k ), for k, we also get S, Z,. {k<c}. (3) Sice the distributio of the out-degree of specified odes i recursive trees Z, has bee studied i [] for the whole rage < oe immediately obtais correspodig results for S, ad I, also, which are give ext. Corollary 5. The distributio of Z,, S, ad I, are give as follows: ( P{S, m} P{Z, m} P{I, m} ) km ( )[ k k 2 m], for <. 2 k! (4) The limitig distributio behavior of the radom variable X,, which stads for Z,, S, or I,, is, for ad depedig o the growth of, give as follows. The regio for small: o(). The suitably scaled ad shifted r. v. X, is asymptotically Gaussia distributed, X, : X, (log log ) log log N (0, ). (5) The cetral regio for : such that µ, with 0 < µ <. The radom variable X, is asymptotically Poisso distributed Poisso(λ) with parameter λ log µ. X, X µ, with P{X µ m} The regio for large: l : o(). P{X, 0}. We also give the followig corollary. µ( log µ)m. (6) m! 6

7 Corollary 6. The umber I,,m of permutatios of {, 2,..., }, where exactly m elemets have iversios is give by ( )[ k k I,,m!( )! m] k!. (7) km I,,m is for ad m fixed ad asymptotically give by I,,m 2π m! 3 2 log m () e, (8) whereas for l : ad m fixed ad by 2π m I,,m m!(l m)! e. (9) Remark 2. Note that, by usig the correspodece betwee recursive trees ad permutatios as give by Biectio ad results stated i [2] about the ode degrees i recursive trees, oe ca get eve a refiemet of the results preseted. It is possible to obtai a closed formula for I,J,M, which gives the umber of permutatios of {, 2,..., }, where exactly m i elemets have i iversios, with i r, for vectors J (,..., r ) ad M (m,..., m r ). 3.2 The sum of the labels of odes attached via edges of a give weight We study ow the radom variable W, : e(e,e 2) E e 2 {we}, which couts the sum of the labels of odes attached via edges of weight, where agai E deotes the set of edges of a radom recursive tree of size. I order to get a direct correspodece with iversios we study a variat, i.e., the r. v. W, : e(e,e 2) E (e 2 ) {we}, ad obtai as a cosequece of Biectio the followig propositio. Propositio 2. The radom variable W, couts the sum of the values of the elemets i a radom permutatio of {, 2,..., }, which have exactly iversios. We immediately obtai due to cosideratios aalogous to the proof of Theorem 5 the followig distributio law of W,. Theorem 6. The distributio of the r. v. W, is give as follows. W, B B, (20) where B k deotes a scaled Beroulli distributed r. v. k Be( k ), i.e., P{B k k} /k ad P{B k 0} /k. The ext theorem characterizes the limitig distributio of W, for the three phases appearig depedig o the growth of (). Theorem 7. The limitig distributio behaviour of the r. v. W, is, for ad depedig o the growth of, give as follows. The regio for small: o(). The limitig distributio of the suitably scaled r. v. W, is Dickma s ifiitely divisible distributio, W, x W, with P{W < x} e γ 0 ρ(v)dv, for x > 0, (2) where ρ(v) deotes the Dickma fuctio ad γ is the Euler-Mascheroi costat. 7

8 The cetral regio for : such that µ, with 0 < µ <. The suitably scaled r. v. W, coverges i distributio to a r. v. W µ, whose distributio ca be characterized via its Laplace trasform, W, ( t W µ, with ψ µ (t) : E(e twµ ) exp µt Furthermore the distributio of W µ is ifiitely divisible. The regio for large: l : o(). W, e v ) dv. (22) v 0. (23) For a overview of the Dickma distributio, the Dickma fuctio ad a extesive list of combiatorial obects, which lead to the Dickma distributio, we refer to [7]. Proof. The proof of the case already appeared i [7] ad a extesio of the argumets appearig there leads to the theorem preseted. As a cosequece of Theorem 6 the Laplace trasform ψ, (t) : E(e t W, ) of W, / is give by ψ, (t) k ( + e tk k ). (24) First we cosider the case o(). Usig Curtiss theorem [5] it suffices to show that the Laplace trasform ψ, (t) coverges poitwise i a eighborhood of t 0 to the Laplace trasform of the Dickma distributio, i.e., We write equatio (24) as t lim ψ,(t) 0 ψ, (t) exp ( log ( + k e v dv. (25) v e tk By usig the expasio log( + x) k ( x)l /l we get further ( ψ, (t) exp k e tk k ad by stadard estimates we obtai for the remaider term R, (t) l 2 ( ) l l k (e tk ) l ) ). (26) k ) + R, (t), (27) k l O( t 2 /), (28) uiformly for all,. By a applicatio of the Euler-MacLauri summatio formula ad the estimate (28) we get the followig expasio of the sum appearig i (27): k e tk k t( ) t e v dv + O( t 2 /). (29) v This leads to the stated result for o(). 8

9 For the case µ, with 0 < µ <, we proceed as before ad get the followig expasio of the sum appearig i (27): k e tk k t( ) µt e v dv + O( t 2 /), (30) v which agai proves the stated result. The ifiitely divisibility follows by usig the characterizatio of Chow ad Teicher [4], p The remaiig case l : o() is prove fully aalogous. Remark 3. It ca be see easily that the limit law of W, is the same as the limit law of W,, ad thus Theorem 7 also holds for W,. 4 Weighted depths ad distaces 4. The edge-weighted depth of odes ad distace betwee odes A weighted recursive tree readily provides geeralizatios to depths ad distaces by addig the edge-weights o the coectig path betwee two specified odes istead of simple coutig the umber of edges. We start with a simple observatio leadig to the (degeerate) distributio of the edge-weighted depth X,, which is give by the sum of the weights of the edges lyig o the uique path from the root to ode i a radom recursive tree of size. Propositio 3. The r. v. X, satisfies the followig distributio. X, X,. (3) Proof. Cosider a recursive tree of size ad let us assume that the coectig path from the root to ode is give by the ode sequece v 0, v,..., v r. The the sum of the edge-weights o this path is give by i r (v i v i ) v r v 0, which proves this propositio. Next we are goig to study the edge-weighted distace betwee arbitrary odes 2 i radom recursive trees of size, where it is sufficiet to cosider oly the case 2, sice odes larger tha 2 do ot have a ifluece o this parameter (this is a cosequece of the descriptio of radom recursive trees via a tree evolutio process). To do this we itroduce the r. v. X,, which gives the sum of the weights of the edges lyig o the uique path from ode to ode i a radom recursive tree of size. The ext propositio reduces the aalysis of X, to the special istace +. Propositio 4. X, ( ) X +,. (32) Proof. A combiatorial argumet is the followig. The edge-weighted distace betwee the odes ad i a size- recursive tree depeds oly o the label of the root of the spaig tree of ad. Assume that ode k, with k, is the root of the spaig tree of the odes ad. The the edge-weighted distace is give by + 2k, regardless of the odes lyig o the path from k to. Now merge all odes o the path from k to with labels larger tha ito oe ode ad delete all other odes larger tha except itself. Thik of the ew ode as ode +. Thus to obtai the edge-weighted distace betwee ad we oly have to add to the edge-weighted distace betwee odes ad +, which proves the stated result. A more probabilistic argumet is also give ext. P{X, m} P{X, m < c }P{ < c } + P{X, m c }P{ c } P{X, m }P{ < c } + P{X, m }P{ c } 9

10 P{X, m }. (33) Iteratig this argumet leads to the stated result. Remark 4. Propositio 3 ad Propositio 4 are valid for a larger class of trees, i.e., for the family of so called grow simple icreasig trees, see [8, ] for a defiitio of this tree family. The distributio of X +, is characterized ext. Theorem 8. The r. v. X +, satisfies the followig distributio law. { P{X +, 2m }, m, (+2 m)(+ m), < m. (34) The expectatio ad the variace of X +, are give by the followig exact formulæ. E(X +, ) 2 2H +, V(X +, ) 8 4H 2 4H. (35) The limitig distributio of the suitably shifted r. v. X +, is give as follows. X +, 2 2R, with P{R m} Thus the iteger momets of R do ot exist., for m N. (36) m(m + ) Proof. We start with the remark that, as a cosequece of the proof of Propositio 4, X +, ca have oly values, 3, 5,..., 2 3, 2. Coditioig o the ode l, where + is attached, gives the P{X +, 2m } [[m ]]P{ + < c } + P{X +, 2m + < c l}p{ + < c l} [[m ]] [[m ]] [[m ]] Takig differeces leads the to + l P{X,l 2m ( + l) + < c l} l P{X,l 2m ( + l)} + l P{X l+,l 2m (2 2l)}. (37) + l P{X +, 2m } ( )P{X, 2m 3} [[m ]] [[m 2]] + P{X, 2m 3}, ad after ormalizatio we obtai the recurrece Iteratig (38) gives P{X +, 2m } P{X, 2m 3} + P{X +, 2m } k [[m k]] [[m k + ]] + k [[m ]] [[m 2]]. (38) + P{X 2, 2m 2( )}, (39) ad after simplifyig the expressio we get the stated distributio law. The formulæ for the expectatio ad the variace are obtaied by easy summatio. To obtai the limitig distributio result we simply observe that { m(m+) P{X +, 2 2m}, m <,, m. (40) 0

11 4.2 The root of the spaig tree of cosecutive odes Let us deote by R +, the radom variable, which couts the label of the root of the spaig tree of the odes ad + i a radom recursive tree of size +. Usig our results for X +, as obtaied i Subsectio 4. we immediately obtai the followig theorem. Theorem 9. The r. v. R +, satisfies the followig distributio law. { m(m+) P{R +, m}, m <,, m. (4) The expectatio ad the variace of R +, are give as follows. E(R +, ) H, V(X +, ) 2 2H. (42) Furthermore R +, coverges i distributio to a r. v. R, without covergece of ay iteger momet, R +, R, with P{R m}, for m N, (43) m(m + ) Proof. As remarked i the proof of Propositio 4 the weighted distace betwee odes ad is give by the smallest label o the coectig path of ad ad thus by the root of the spaig tree. We obtai that the weighted distace is + 2m if ad oly if ode m is the root of the spaig tree. Thus P{R +, m} P{X +, 2( + m) }, ad the exact distributio of R +, is characterized, which immediately also leads to the limitig distributio result. Agai the formulæ for the expectatio ad the variace are obtaied easily. Remark 5. Note that curiously R +, is distributed as the size of the subtree rooted at a radomly chose ode i a radom recursive tree of size, [0]. 4.3 The root of the spaig tree of two radomly chose odes We study also the radom variable Y [R], which gives the label of the root of the spaig tree of two radomly chose odes i a radom recursive tree of size. Corollary 7. The radom variable Y [R] momet. P{Y [R] + m} m(m + ) + 2 m(m + ), for m. (44) ( + ) Y [R] coverges i distributio to a r. v. Y, without covergece of ay iteger Y, with P{Y m}, for m N. (45) m(m + ) Proof. Sice two odes i a recursive tree of size + are chose at radom, ay pair (i, l), with 2 i < l +, is selected with probability (+). Thus we obtai P{Y [R] + m} m i<l + m i<l + 2 ( + ) P{X l,i i + l 2m} 2 ( + ) P{X i+,i 2i 2m + } 2 ( + m ( + ) m + 2 ( + ) im+ ( + i)p{x i+,i 2i 2m + } im ) ( + i)p{x i+,i 2i 2m + }. (46) Sice P{X i+,i 2i 2m + } m(m+), for m < i, due to Theorem 8, we obtai the stated exact distributio result by summatio ad the limitig distributio result easily follows from that.

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