Subtrees of a radom tree Bogumi l Kamiński Pawe l Pra lat November 21, 2018 Abstract Let T be a radom tree take uiformly at radom from the family of labelled trees o vertices. I this ote, we provide bouds for c), the umber of sub-trees of T that hold asymptotically almost surely a.a.s.). With computer support we show that a.a.s. 1.41805386 c) 1.41959881. Moreover, there is a strog idicatio that, i fact, a.a.s. c) 1.41806183. 1 Itroductio I this paper, we are cocered with the problem of fidig bouds for the umber of sub-trees of a radom tree o vertices. Clearly, the path P ad, respectively, the star K 1, 1 have the most ad the least sub-trees amog all trees of order. The biary trees that maximize or miimize the umber of sub-trees are characterized i Székely ad Wag 2005, 2007). There is a uexpected coectio betwee the biary trees which maximize the umber of subtrees ad the biary trees which miimize the Wieer idex, a chemical idex widely used i biochemistry; the the Wieer idex is defied as the sum of all pairwise distaces betwee vertices Wieer 1947). Subtrees of trees with give order ad maximum vertex degree are studied i Kirk ad Wag 2008). The extremal trees coicide with the oes for the Wieer idex as well. Fially, trees with give order ad give degree distributio was cosidered i Zhag et al. 2013). I this paper, we ivestigate c), the umber of subtrees of a radom tree T take uiformly at radom from the family of labelled trees o vertices. The tree T is called a radom tree or radom Cayley tree). The classical approach to the study of the properties of T was purely combiatorial, that is, via coutig trees with certai properties. I this way, Réyi ad Szekeres, usig complex aalysis, ivestigated the height of T. Perhaps surprisigly, it turs out that the typical height is of order Réyi ad Szekeres 1967). Now, a useful relatioship betwee certai characteristics of radom trees ad brachig processes is established. I fact, recetly ad idepedetly of this work, Cai ad Jaso 2018) ivestigated the umber of subtrees i a coditioed Galto Watso tree of size. They, i particular, showed that logc)) has a Cetral Limit Law ad that the momets of c) are of expoetial scale. Moreover, i a earlier work, Wager 2012) used these techiques to show that logc)) is asymptotically ormally distributed, with mea ad variace asymptotically equal to µ ad σ 2 respectively, where the umerical values of µ ad σ 2 are µ 0.35 slightly less tha 1/e; e µ 1.419067549) ad σ 2 0.04. I this paper, istead of exploitig this probabilistic poit of view we approach the problem through combiatorial perspective which, presumably, gives stroger asymptotic bouds for SGH Warsaw School of Ecoomics, Warsaw, Polad Departmet of Mathematics, Ryerso Uiversity, Toroto, ON, Caada. 1
c). Moreover, this approach allows us to provide bouds for c) that hold asymptotically almost surely a.a.s.), istead of bouds for Ec)). For more o radom trees see, for example, Frieze ad Karoński 2015) or Lyos ad Peres 2016). Our mai results are preseted i Sectio 2. After itroducig the otatio we move to a lower boud that does ot require computer support; see Sectio 2.4. The strogest lower boud, with support of a computer, is preseted i Sectio 2.5 culmiatig with Theorem 2.5 which gives that a.a.s. c) 1.41805. The strogest upper boud ca be foud i Sectio 2.7; Theorem 2.9 implies that a.a.s. c) 1.41960. I the fial sectio of the paper, Sectio 3, we preset a cojecture that we are rather cofidet is true) that would determie the first 5 digits of c); see Cojecture 3.1. There is also a short discussio of the outcome of applyig the geeral result of Zhag et al. 2013) o the umber of subtrees of a tree with a give order ad degree distributio. The fial subsectio discusses briefly complemetary simulatios that we performed durig this project. The umerical results preseted i this paper i particular, Table 1) were obtaied usig Julia laguage Bezaso et al. 2017). The computatios were performed o AWS EC2 takig i total approximately 1,000 hours of computig. 2 Theoretical bouds 2.1 Asymptotic otatio Each time we refer to T i this paper, we cosider a labelled tree o the vertex set [] take uiformly at radom from the set of all labelled trees o vertices. As typical i radom graph theory, we shall cosider oly asymptotic properties of T as. We emphasize that the otatios o ) ad O ) refer to fuctios of, ot ecessarily positive, whose growth is bouded. We use the otatios f g for f og) ad f g for g of). We also write f) g) if f)/g) 1 as that is, whe f) 1 + o1))g)). We say that a evet i a probability space holds asymptotically almost surely a.a.s.) if its probability teds to oe as goes to ifiity. 2.2 Prüfer code Let us start with recallig a classic result that will be useful i our aalysis. Prüfer code of a labelled tree T o vertices is a uique sequece from [] 2 the set of sequeces of legth 2, each term is from the set [] {1, 2,..., }) associated with tree T Prüfer 1918). I fact, there exists a bijectio from the family of labelled trees o vertices ad the set [] 2. This, i particular, implies that the Cayley s formula holds: the umber of labelled trees o vertices is 2. More importatly, it gives us a way to geerate a radom labelled tree by simply selectig a radom elemet from [] 2 ad cosiderig the correspodig tree T. Suppose a labelled tree T has a vertex set []. Oe ca geerate a Prüfer code of T by iteratively removig vertices from the tree util oly two vertices remai. At step i of this process, remove the leaf with the smallest label ad set the ith elemet of the Prüfer code to be the label of this leaf s eighbour. 2
2.3 Lower boud: trivial approach Cosider the Prüfer code of T. Clearly, the degree of ay vertex v is the umber of times v appears i the code plus 1. It follows that for ay v [] V T ) ad ay k N, P degv) k) 2 k 1 ) 1 ) k 1 1 1 ) k 1 e 1 k 1)!. 1) Now, let X 1 be the umber of leaves of T. From above it follows that E[X 1 ] /e ad we ca easily prove usig, say, the secod momet method) that a.a.s. X 1 /e. We will prove a more geeral result below see Lemma 2.2 so we skip a formal argumet here.) Oe ca select all o-leaves ad the ay subset of the leaves to form a sub-tree. Note that ay subset of leaves ca be safely removed ad so ay choice results with a coected graph.) We get the followig lower boud that holds a.a.s.: c) 2 X1 2 1/e+o1)) 2 1/e + o1)) 1.29045. 2.4 Lower boud: warmig up o a piece of paper... The reaso for this sectio is twofold. First of all, we preset a lower boud that does ot require computer support. Aother reaso is to prepare the reader for a more sophisticated argumet preseted i the ext sectio that will give a stroger boud but will require computer support. Theorem 2.1. A.a.s. c) 1.37135. Proof. Let γ be a sufficietly large iteger that will be determied soo. For k {2, 3,..., γ}, let X k be the umber of subsets S [] of size k that iduce a star K 1,k 1 ) ad the oly edge coectig S to the rest of T is adjacet to the ceter of the star. I particular, the k 1 leaves of the star are leaves i T. Trivial, but importat, property is that vertices of T that belog to K 1,k 1 caot be part of some other K 1,k 1 for some k that could be equal to k but does ot have to be). We put vertex v of T together with the k 1 leaves adjacet to v) ito C k if v belogs to some K 1,k 1. As a result, we partitio the vertex set ito a family of classes C k k {2, 3,..., γ}; C k cotais X k stars ad so it cotais X k k vertices), leaves L that are ot part of ay earlier class, ad R that cotais the remaiig vertices of T. By cosiderig a radom Prüfer code, we get that a.a.s., for ay k {2, 3,..., γ} X k ) ) k 1 1 k 1 k ) k 1 e k k k 1)! ; there are k) choices for S, k choices for the root, each leaf selects the root with probability 1/, with probability 1 k/) k 1 o vertex picked leaves ad o vertex other tha the leaves picked the root. More geeral result will be proved i the ext subsectio see Lemma 2.2.) The umber of leaves i L is a.a.s. ) γ γ L X 1 X k k 1) e 1 e k k 1) β L, k 1)! where β L β L γ) is the costat that ca be made arbitrarily close to ˆβ L : e 1 k 2 e k k 2)! e 1 e 1/e 2 0.1724 3
by takig γ large eough. The umber of rooted sub-trees of K 1,k 1 icludig the empty tree) is clearly 2 k 1 + 1. Hece, we get the followig lower boud for c) by takig all vertices of R, ay subset of L, ad ay rooted sub-trees from classes C k : a.a.s. γ c) 2 L 2 k 1 + 1 ) X k 2 β L+o1) 2 β L γ 2 k 1 + 1) e k /k 1)!+o1) γ 2 k 1 + 1) e k /k 1)! + o1)) β + o1)), where β βγ) is the costat that ca be made arbitrarily close to ˆβ : 2 ˆβ L 2 k 1 + 1) e k /k 1)! 2 e 1 e 1/e 2 2 k 1 + 1) e k /k 1)! > 1.37135 k 2 by takig γ large eough. The desired boud holds. k 2 2.5 Lower boud: computer assisted argumet I this sectio, we geeralize the strategy we cosidered i the previous sectio. Istead of restrictig ourselves to stars, we ivestigate all possible trees o k vertices, where k K for some value of K. Ufortuately, it seems impossible to fid a closed formula for the umber of trees with a give umber of sub-trees but, with computer assist, we ca do it eve for relatively large values of K. As before, oe could iclude a arbitrarily large) family of stars but this improvemet is egligible ad so we do ot do it. Fix some K N. We start with a few importat defiitios. ) Family F k For each k [K], let F k be the family of rooted trees o k vertices; that is, each member of F k is a pair T, v), where T is a labelled tree o the vertex set [k] ad v [k]. Clearly, F k k k 2 k k k 1. Fially, let F K k1 F k. Vertices of type T, v) ad iteral vertices For each vertex v of T, we cosider l degv) sub-trees of T T 1, T 2,..., T l ), all of them rooted at v, that are obtaied by removig oe of the l edges adjacet to v. Now, each T i o k i vertices) is re-labelled so that labels are from [k i ] but the relative order is preserved. Sice we aim for asymptotic results, we may assume that > 2K ad so at most oe such rooted tree, say T 1, v), belogs to F. If this is the case, the we say that v is of type T 1, v) ad that it iduces rooted tree T 1, v); otherwise, we say that v is a iteral vertex. Partitio of the vertex set of T We partitio the vertex set of T set []) as follows. We start the process at roud K. It will be coveiet to cout rouds from K dow to 1.) For each vertex of type T, v), for some T, v) F K, we put all the vertices of the rooted sub-tree it iduces ito class CT, v). Note that o vertex of T belogs to more tha oe sub-tree as we cosider oly types from F K trees of a fixed size). Hece, i particular, the classes created so far are mutually disjoit. O the other had, all vertices of type differet tha T, v) that are placed ito class CT, v) are of type from F \ F K. Hece, i order to avoid placig oe vertex ito more tha oe class, we eed to trim the tree ad remove all vertices that are already placed ito some class. Roud K is fiished ad ow we move to the ext roud, 4
roud K 1, i which vertices of types from F K 1 are cosidered ad proceed the same way. Note that ot all of them are removed durig roud K.) We do it recursively all the way dow to roud 1 durig which F 1 is cosidered ad so the remaiig leaves of T are trimmed. The oly vertices left are iteral oe which are placed ito set R. We obtai the followig partitio of []: {CT, v) : T, v) F} {R}. We start with estimatig the umber of vertices of each type. Lemma 2.2. For ay K N, the followig property holds a.a.s. For ay T, v) F k for some k [K], the umber of vertices of type T, v) is 1 + o1))e k /k!. Proof. The argumet is a straightforward applicatio of the secod momet method. Fix ay k [K] ad T, v) F k ; we will show that the desired boud holds a.a.s. for this choice. This will fiish the proof as the umber of choices for k ad T, v) is bouded ad so the coclusio holds by the uio boud. For ay S [], S k, let IS) be the idicator radom variable that set S iduces a tree T rooted at v after relabellig preservig the order of vertices of S) ad the oly edge from S to its complemet is adjacet to a vertex re-labelled as v. The umber of vertices of type T, v) is X IS). For ay S we have p : PIS) 1) S [], S k ) k 1 1 1 k k 1 ) k 1) e k. Ideed, without loss of geerality, we may assume that S {1, 2,..., k}. The, the first k 1 terms of the Prüfer code of T are completely determied by T ad v hece term 1/) k 1 ); moreover, the remaiig 2) k 1) k 1 terms caot be from S hece term 1 k/) k 1 ). It follows that E[X] ) p k p k k! e k. k! Now, Var[x] Var IS) S,S ) + S S [], S k PIS) 1, IS ) 1) PIS) 1) 2) PIS) 1) PIS) 1) 2 ), where ) meas that the sum is take over all pairs of sets S, S [] with S S k. The secod term i the last sum ca be dropped to get ad upper boud of E[X] for the last sum. More importatly, ote that if S ad S itersect, the PIS) 1, IS ) 1) 0. Hece, Var[x] PIS) 1, IS ) 1) PIS) 1) 2) + E[X], S,S ) 5
where ) meas that the sum is take over all pairs of disjoit sets S, S [] with S S k. For ay such pair, q : PIS) 1, IS ) 1) PIS) 1) 2 ) 2k 1) 1 1 2k ) 2) 2k 1) ) k 1 1 1 k ) ) 2) k 1) 2 ) 2k 1) 1 1 2k ) 2k 1 k ) ) 2 2k 2. Usig the fact that 1 x exp x x 2 /2 + Ox 3 )) ad the that expx) 1 + x + Ox 2 ), we get ) )) q : exp 2k 2k 1) + 2k2 + O 2 ) exp 2k + k2 + 2k + O 2 ) k 2k 1) e 1 2k 2 )) + 2k exp + O 2 ) p2 k 2 2k). It follows that Var[x] k ) k k ) q + E[X] ) ) 2 k 2 2k p + E[X] oe[x] 2 ). k The secod momet method implies that a.a.s. X E[X] ad the proof is fiished. Now, we are ready to aalyze the trimmig process that yields the desired partitio of the vertex set of T. Lemma 2.3. For ay K N, the followig property holds a.a.s. For ay T, v) F k for some k [K], CT, v) k f K k), where f K k) : e k k! K ) l e l k) l k 1 l l k l!. lk+1 Proof. Sice we aim for a statemet that holds a.a.s., we may assume that T is ay labelled tree o the vertex set [] that satisfies properties stated i Lemma 2.2. The desired property will hold determiistically. To that ed, we eed to aalyze the trimmig process. Fix ay T, v) F K. Durig the first roud that is, roud K), all vertices of type T, v), together with the correspodig trees that are iduced by them, are moved to class CT, v). By Lemma 2.2, the umber of vertices of type T, v) is 1 + o1))e K /K! ad so CT, v) / K ˆf K K), where ˆf K K) : e K K!. Now, cosider ay roud k 1 k < K) ad suppose that the process is already aalyzed up to that poit; that is, durig rouds l k+1 l K), for ay T, v) F l, 1+o1)) ˆf K l) vertices of type T, v) were moved to class T, v) as usual, together with the correspodig trees that are iduced by them). Fix ay T, v) F k. By Lemma 2.2, at the begiig of the trimmig process there were 1 + o1))e k /k! vertices of type T, v). Some of them were trimmed durig some roud l k + 1 l K); but how may of them? I order to aswer this questio we eed to kow how may rooted trees o l vertices cotai a vertex of type T, v). We are goig to use a argumet similar to the oe used i the proof of Lemma 2.2. There are l k) ways to select labels for the sub-tree o k vertices of 6
a tree o l vertices. Without loss of geerality, we may assume that the selected labels are {1, 2,..., k}. Now, the Prüfer code for a super-tree o l vertices has to have the first k 1 terms as determied by T ad v. The remaiig l 2) k 1) l k 1 terms yield all possible super-trees; each of these terms is from [l] \ [k]. Sice there are l k) choices for the root of a tree o l vertices, we get that the aswer to our questio is l k) l k 1 l k) ) l k l k) l k l k). It follows that the umber of vertices of type T, v) that survived till roud k is 1 + o1)) ˆf K k), where ˆf K k) : e k k! K lk+1 ) l l k) l k ˆf K l), 2) l k ad so CT, v) / k ˆf K k). It remais to show that f K k) ˆf K k) for 1 k K; we prove it by strog iductio o k. Clearly, f K K) ˆf K K) so the base case holds. Suppose the that f K l) ˆf K l) for k + 1 l K ad our goal is to show that f K k) ˆf K k). From this ad 2) we get ˆf K k) e k k! e k k! K lk+1 K lk+1 ) l l k) l k f K l) l k ) l l k) l k e l l k l! K ml+1 ) ) m e m l) m l 1 m. m l m! We will show that the terms i ˆf K k) cotaiig e a for k < a K are the same as the oes i f K k). Clearly, it is the case for a k.) To see this, ote that oe of these terms is preset i the above equatio for l a see the first part iside the parethesis) ad oe for each k < l < a see the term correspodig to m a i the secod part iside the parethesis). Collectig those terms i ˆf K k) we get: ) a e a k) a k a + a k a! a 1 lk+1 ) l l k) l k l k a l) a l 1 a a l ) e a a!. O the other had, the oly term i f K k) cotaiig e a is a k) a k 1 a a k) e a /a!. Hece, to fiish the iductive step it is eough to show that ) a a k 1)a k) a k 1 a k a 1 lk+1 ) l l k) l k l k which, after substitutig b a k ad c l k, we ca rewrite as c1 c1 a l) a l 1 a a l b 1 ) b b 1 ) b b 1)b b 1 c c b c) b c 1 c c c 1 b c) b c 1. 3) c c The, by settig d b c i the first step, ad the usig the fact that b b d) b d) ad chagig d to c i the otatio i the secod step, we get b 1)b b 1 ), b 1 ) b b d) b d) b d 1 d d 1 b d d1 b 1 ) b b c) b c) b c 1 c c 1. 4) c c1 7
By addig 3) ad 4) ad dividig both sides by b we get b 1 ) b 2b 1)b b 2 c c 1 b c) b c 1. 5) c c1 The fial puzzle piece missig is the proof of 5) for which we will use a bijective argumet. The left had side of 5) couts all labelled trees o the vertex set [b] with oe edge selected ad orieted. Now, cosider the followig costructio. First, take ay proper ad o-empty subset C [b] of size c 1 c b 1); let D [b] \ C. Costruct ay labelled tree o C ad select oe vertex v C C. Similarly, costruct ay labelled tree o D ad select oe vertex v D D. Fially, coect v C to v D by a orieted edge from v C to v D. The described costructio geerates all possible labelled trees with oe edge selected ad orieted. Moreover, each such tree is costructed exactly oce. Now observe that umber of such costructios is equal to right had side of 5). The proof is fiished. Now, we are ready to state the mai result of this sub-sectio that yields the strogest lower boud we have. Theorem 2.4. Fix ay K N. For ay T, v) F k for some k [K], let gt, v) be the umber of sub-trees of T cotaiig v. Let f K k) be defied as i the statemet of Lemma 2.3. The, the followig boud holds a.a.s. K c) gt, v) fkk) + o1). 6) k1 T,v) F k Proof. Recall that the vertex set of T is partitioed as follows: for T, v) F, set CT, v) cotais vertices of type T, v) that iduce rooted trees T, together with other vertices of T ; the iteral vertices form set R. It follows from Lemma 2.3 that a.a.s., for ay k [K] ad ay T, v) F k, the umber of rooted trees i CT, v) is 1 + o1))f K k). By takig all vertices of R ad ay rooted sub-trees from CT, v), the followig lower boud for c) holds: a.a.s. K K c) gt, v) fkk)+o1) gt, v) fkk) + o1), k1 T,v) F k k1 T,v) F k sice the umber of terms i this product is bouded. Fuctio f K k) ca be easily calculated umerically) eve for relatively large values of K ad k. Ufortuately, there is o closed formula for gt, v), the umber of rooted sub-trees of T recall that the empty tree is icluded). O the other had, gt, v) ca be easily computed with computer support usig the followig simple, recursive algorithm. Let Nv) be the set of eighbours of v. For ay w Nv), T vw that is, forest obtaied after removig edge vw) cosists of two sub-trees; let ST, v, w) be the sub-tree cotaiig w. The gt, v) ca be computed as follows: if T is K 1 isolated vertex), the gt, v) 2; otherwise, gt, v) 1 + gst, v, w), w). 7) w Nv) Actual computatios of c) ca be made efficiet usig the followig two observatios: 8
1. we do ot have to explicitly geerate all trees T, v) i F k ; it is eough to cout the umber of rooted trees of size k that have a give value of gt, v) sice this is eough to compute 6); 2. if we start from k 1 up to k K, the we ca derive couts of trees from F k with uique values of gt, v) usig couts of umbers of trees from F k s, where s [k 1], with uique values of gt, v) as i 7), the right had side cosiders trees of size oe less tha the left had side. The exact procedure is give i Algorithm 1, where xk, g) {T, v) F k : gt, v) g}. Usig xk, g), oe ca rewrite 6) as follows: K c) xk,g) g f Kk) K + o1) f K k) g xk,g) + o1), g N g N k1 k1 Algorithm 1 Algorithm for calculatio of xk, g). k, g N : xk, g) 0 x1, 2) 1 for k {2, 3,..., K} do for all a 1, a 2,..., a m N such that m i1 a i k 1 ad a i a i+1 do let j, j [p], be the legth of the j-th costat subsequece of the a i sequece for all xa i, g i ) over all i [m] ad g i N do xk, 1 + m i1 g i) xk, 1 + m i1 g k! i) + m p j1 j! i1 ed for ed for ed for xa i,g i) a i! The obtaied lower bouds for K 1, 2,..., 30 are preseted i Table 1 colum lower, the followig colums are explaied i the followig sectios). Clearly, the strogest boud is yielded by K 30 ad is the best lower boud we have. Theorem 2.5. A.a.s. c) 1.41805. 2.6 Upper boud: trivial approach Recall that i the proof of Theorem 2.1 we partitio the vertex set of T ito a family of classes C k k {2, 3,..., γ}; C k cotais X k stars ad so it cotais X k k vertices), leaves L that are ot part of ay earlier class, ad R that cotais the remaiig vertices of T. The size of L is already estimated i Theorem 2.1. The umber of vertices that belog to some class C k is a.a.s. γ γ C k X k k γ e k k 1)! k β C, where β C β C γ) is the costat that ca be made arbitrarily close to ˆβ C : k 2 e k k 1)! k e1/e 1 + e 1/e 2 e 1 0.3591 by takig γ large eough. Fially, a.a.s. R Ck L 1 βc β L ) β R, 9
where β R β R γ) 1 β C β L is the costat tedig to ˆβ R : 1 ˆβ C ˆβ L 1 e 1/e 1 0.4685 as γ. To get a upper boud for c), we select ay subset of R L ad ay rooted sub-trees from classes C k. Clearly, each sub-tree of T is achieved but ot all selected sets iduce a coected graph. I fact, almost all of them do ot!) So we are clearly over-coutig but the followig ca serve as the upper boud that holds a.a.s.: γ c) 2 L + R 2 k 1 + 1 ) ) X k γ 2 β L+β R +o1) 2 k 1 + 1) e k /k 1)!+o1) γ 2 β L+β R 2 k 1 + 1) e k /k 1)! + o1)) α + o1)), where α αγ) is the costat that ca be made arbitrarily close to ˆα : 2 ˆβ L + ˆβ R 2 k 1 + 1) e k /k 1)! k 2 2 1+e 1 e 1/e 1 e 1/e 2 2 k 1 + 1) e k /k 1)! < 1.89756 k 2 by takig γ large eough. It follows that a.a.s. c) 1.89756. The same trivial argumet ca be used to adjust Theorem 2.4: the ratio betwee the upper ad the lower boud is 2 R, where R is the set of iteral vertices. The followig straightforward corollary of Lemma 2.2 estimates the size of R. It shows that the fractio of vertices that are iteral is tedig to zero as K. This is, of course, a desired property as it implies that the gap betwee the upper ad the lower boud for c) ca be made arbitrarily small by cosiderig large values of K. Ufortuately, the rate of covergece is ot so fast. Corollary 2.6. For ay K N, a.a.s. R hk) : 1 K where a asymptotic is with respect to K. k1 k/e) k k k! ) 1 Θ, K Proof. The umber of iteral vertices that is, vertices that are ot of type T, v) for ay T, v) F) ca be estimated usig Lemma 2.2. Sice F k k k 1, we get that R 1 K k1 F k e k k! 1 K k1 k/e) k k k! hk). To see the secod part, cosider the brachig process i which every idividual produces idividuals that is a idepedet radom variable with Poisso distributio with expectatio 1. The process exticts with precisely k idividuals i total) with probability k/e)k k k! see, for example, Taer 1961)). Hece, hk) is the probability that the total umber of idividuals is more tha K. Sice the process exticts with probability 1, hk) 0 + as K ; or, alteratively, k/e) k k 1 k k! 1. To see the rate of covergece we apply Stirlig s formula k! 2πkk/e) k to get hk) ) k/e) k ) 1 Θ k 3/2 Θ. k k! k>k k>k K The proof is fiished. 10
We get the followig couterpart of Theorem 2.4. Observatio 2.7. Fix ay K N. For ay T, v) F k for some k [K], let gt, v) be the umber of sub-trees of T cotaiig v. Let f K k) ad hk) be defied as i the statemets of Lemma 2.3 ad Corollary 2.6, respectively. The, the followig boud holds a.a.s. K c) 2 hk) gt, v) fkk) + o1). k1 T,v) F k The umerical values of the upper bouds for c) ad for R / K {1, 2,..., 30}) are preseted i Table 1 see colum upper 1 ad colum R /, respectively). For K 30 we get that a.a.s. c) 1.56727. As already metioed, ufortuately, the rate of covergece is ot so fast. Sice the computatioal complexity of the problem makes K to be ot so large at most 30), the umber of iteral vertices is substatial R 0.14434 for K 30) ad so more sophisticated argumets will be eeded. 2.7 Upper Boud: computer assisted argumet We cotiue usig the otatio ad defiitios used i Sectio 2.5. Recall that the vertex set of T is partitioed there as follows: for T, v) F, set CT, v) cotais vertices of type T, v) that iduce rooted trees T, together with other vertices of T ; the iteral vertices form set R. However, this time we additioally partitio R ito two sets: R L cotais vertices of type T, v) F k for some K < k ˆK light iteral vertices) ad R H R \ R L heavy iteral vertices). Here is the strogest upper boud we have, i its geeral form. Theorem 2.8. Fix ay K, ˆK N such that K < ˆK. For ay T, v) F k for some k [K], let gt, v) be the umber of sub-trees of T cotaiig v. Let f K k) ad hk) be defied as i the statemets of Lemma 2.3 ad Corollary 2.6, respectively. The, the followig boud holds a.a.s. where c) ξ 1 ξ 2 ξ 3 ξ 4 + o1)), ξ 1 ξ 2 ξ 3 ξ 4 ) h ˆK) ˆK + 1 ˆK ˆK k + 1 kk+1 ˆK kk+1 K k1 k ) kk 1) k 2 e k /k! ) k 2k 1 k 1 kk 1) k 2 )e k /k! 2k 2 T,v) F k gt, v) fkk). Proof. Let us fix ay vertex r []. Our goal is to use 7) to estimate gt, r), the umber of sub-trees of T cotaiig r. As metioed earlier, [] is partitioed ito sets CT, v) cotaiig trees rooted at vertices of type T, v), R L ad R H cosistig of light ad, respectively, heavy iteral vertices. It follows from Lemma 2.3 that a.a.s., for ay k [K] 11
ad ay T, v) F k, the umber of rooted trees i CT, v) is 1 + o1))f K k). From Corollary 2.6 we get that a.a.s. the umber of heavy iteral vertices is 1 + o1))h ˆK). Fially, Lemma 2.2 implies that a.a.s. for ay T, v) F k for some k [ ˆK], the umber of vertices of type T, v) is 1 + o1))e k /k!. Recall that for ay w Nv), T vw cosists of two sub-trees; ST, v, w) is the sub-tree cotaiig w. The, gt, r) 1 + gst, r, w), w) w Nr) ad gt, v) ca be recursively) computed as follows: if T, v) F k [K] F k, the gt, v) is already kow that is, computed by computer); otherwise, gt, v) 1 + gst, v, w), w) mt, v) gst, v, w), w), where w Nv) w Nv) mt, v) 1 + w Nv) gst, v, w), w) gt, v) w Nv) gst, v, w), w) gt, v) 1. Clearly, for ay T, v) F k we have the followig trivial upper boud: mt, v) k + 1)/k; this boud is sharp as gt, v) k + 1)/k for a rooted path o k vertices. We will use this boud for all pairs T, v) where v is a leaf i T. The umber of pairs T, v) i F k where v is a leaf of T is kk 1) k 2 there are k choices for the label of v, ad k 1) k 2 rooted trees i F k 1 that ca be attached to v to form T ). This explais the term ξ 2. For heavy iteral vertices, we use eve a weaker boud: mt, v) ˆK + 1)/ ˆK. This justifies the term ξ 1. To make our boud stroger, we will use a better estimatio for mt, v) whe v has degree at least 2 i T ad correspods to a light iteral vertex i T. Ideed, if this is the case, the gt, v) 2k 1; this boud is also sharp as gt, v) 2k 1) + 1 2k 1 for a rooted path o k 1 vertices with a leaf attached to the root that is, a path o k vertices rooted at a vertex adjacet to a leaf). Hece, for pairs of this type we have mt, v) 2k 1)/2k 2). The total umber of members of F k is k k 1 ad we already kow how may of them are ot of this type. This justifies the term ξ 3. Puttig all igrediets together we get that a.a.s. gt, r) ξ 1 ξ 2 ξ 3 ξ 4 + o1)), ad so c) ξ 1 ξ 2 ξ 3 ξ 4 + o1)) ξ 1 ξ 2 ξ 3 ξ 4 + o1)) as 1 + Olog /)) 1 + o1)). The proof is fiished. The umerical values of the upper bouds for c) K {1, 2,..., 30} ad ˆK 10, 000) followig from Theorem 2.8 are preseted i Table 1 see colum upper 2 ). Note that i the computatios we are aggregatig very small umbers; therefore, i order to esure umerical soudess of the results, we have performed them usig 1,024 bit matissa ad roudig-up arithmetic. For K 30 we get the followig values ξ 1 < 1.0000008, ξ 2 < 1.0005917, ξ 3 < 1.00049672, ξ 4 < 1.4180539 that lead to the followig upper boud which is the strogest boud we maaged to obtai: Theorem 2.9. A.a.s. c) 1.41960. 3 Coclusios We fiish the paper with a few commets. 12
3.1 Cojecture Let us revisit the proof of Theorem 2.8. It follows that the ratio betwee upper ad lower boud for c) ca be made arbitrarily close to where η : k K+1 T,v) F k mt, v) e k /k! k K+1 mk) : mt, v) T,v) F k mk) kk 1 e k /k!, 1/k k 1 is a geometric mea of mt, v) over all members of F k. We partitioed F k ito two sets to get the two correspodig upper bouds for mt, v) k + 1)/k ad 2k 1)/2k 2)) which yielded costats ξ 2 ad ξ 3. The improvemet after partitioig of F k is rather mild ad the mai reaso for that was to determie the two sigificat digits of c). O the other had, oe ca easily partitio F k ito more sets to improve the upper boud. We do ot follow this approach as the followig, much stroger, property must be true. It is safe to cojecture that mk) is a decreasig fuctio of k ad this is verified to be the case for 1 k K 30 see Table 1 colum multiplier). I fact, it should coverge to zero quite fast so the cojecture is really safe.) Ufortuately, at preset, we caot prove this property; we have tried a umber of coupligs betwee F k+1 ad F k but with o success. If the property holds, the η mk) kk 1 e k /k! mk) kk 1 e k /k! mk) R /. k K+1 k K+1 Usig K 30 ad the umerical value of m30) 1.00003886 we make the followig cojecture. The cojectured bouds implied by smaller values of K ca be foud i Table 1 see colum coj. upper). Cojecture 3.1. A.a.s. c) 1.41806182. I fact, it feels safe to cojecture that the first 5 digits of c) are 1.41805. If the desired property is proved, we would certaily go for k 31 to squeeze the last drop from the argumet. 3.2 Upper boud based o degree distributio Let S π be the family of all trees o vertices with a give o-icreasig degree sequece π d 0, d 1,..., d 1 ). As metioed i the itroductio, it is kow which extremal tree from S π has the largest umber of sub-trees Zhag et al. 2013). This tree, T π, ca be costructed i a greedy way usig the breadth-first search method. First, label the vertex with the largest degree d 0 as v 01. The, label the eighbours of v 01 as v 11, v 12,..., v 1d0 from left to right ad let dv 1i ) d i for i 1,..., d 0. The repeat this for all ewly labelled vertices util all degrees are assiged. As computed i 1), a.a.s. the umber of vertices of degree k is 1 + o1))e 1 /k 1)!. Usig that ad the costructio metioed above, we get that a.a.s. c) 1.52745 which gives a o-trivial boud but is far away from the oe we obtaied. Of course, this is ot too surprisig, ad it cofirms that the degree distributio is ot a crucial factor i our problem; the umber of sub-trees of T is govered by the distributio of small rooted trees from F K k1 F k. 13
Refereces Bezaso, J., Edelma, A., Karpiski, S., ad Shah, V. B. 2017). Julia: A fresh approach to umerical computig. SIAM Review, 59:65 98. Cai, X. ad Jaso, S. 2018). Preprit. No-frige subtrees i coditioed galto-watso trees. Frieze, A. ad Karoński, M. 2015). Itroductio to radom graphs. Cambridge Uiversity Press. Kirk, R. ad Wag, H. 2008). Largest umber of subtrees of trees with a give maximum degree. SIAM J. Discrete Math., 22:985 995. Lyos, R. ad Peres, Y. 2016). Probability o Trees ad Networks, volume 42 of Cambridge Series i Statistical ad Probabilistic Mathematics. Cambridge Uiversity Press, New York. Available at http://pages.iu.edu/~rdlyos/. Prüfer, H. 1918). Neuer beweis eies satzes über permutatioe. Archives of Mathematical Physics, 27:742 744. Réyi, A. ad Szekeres, G. 1967). O the height of trees. Joural Australia Mathematical Society, 7:497 507. Székely, L. ad Wag, H. 2005). O subtrees of trees. Adv. i Appl. Math., 341):138-155. Székely, L. ad Wag, H. 2007). Biary trees with the largest umber of subtrees. Discrete Appl. Math., 1553):374-385. Taer, J. 1961). A derivatio of the borel distributio. Biometrika, 48:222 224. Wager, S. 2012). Additive tree fuctioals with small toll fuctios ad subtrees of radom trees. Discrete Math. Theor. Comput. Sci. Proc. AQ, pages 67 80. Wieer, H. 1947). Structural determiatio of paraffi boilig poits. J. Amer. Chem. Soc., 69:17 20. Zhag, X.-M., Zhag, X.-D., Gray, D., ad Wag, H. 2013). The umber of subtrees of trees with give degree sequece. J. Graph Theory, 733):280 295. 14
K lower upper 1 upper 2 coj. upper R / multiplier 1 1.29045464 2.00000000 1.43208050 2.00000000 0.63212055 2.00000000 2 1.36324560 1.92362926 1.43208050 1.66745319 0.49678527 1.50000000 3 1.39061488 1.86325819 1.43208050 1.55596710 0.42210467 1.30495588 4 1.40310215 1.81740886 1.43138632 1.50231117 0.37326296 1.20085291 5 1.40946163 1.78177319 1.43028338 1.47223973 0.33816949 1.13753267 6 1.41293442 1.75332505 1.42906778 1.45396472 0.31139897 1.09628284 7 1.41492327 1.73007752 1.42789078 1.44231043 0.29011286 1.06831339 8 1.41610182 1.71070328 1.42681559 1.43464704 0.27266454 1.04887463 9 1.41681820 1.69428865 1.42586149 1.42950426 0.25802502 1.03515096 10 1.41726225 1.68018579 1.42502733 1.42600450 0.24551402 1.02536358 11 1.41754178 1.66792305 1.42430332 1.42359935 0.23466147 1.01833777 12 1.41771993 1.65714906 1.42367665 1.42193478 0.22513081 1.01327326 13 1.41783464 1.64759676 1.42313429 1.42077680 0.21667390 1.00961308 14 1.41790914 1.63905962 1.42266416 1.41996815 0.20910325 1.00696374 15 1.41795786 1.63137546 1.42225550 1.41940180 0.20227419 1.00504449 16 1.41798993 1.62441515 1.42189906 1.41900425 0.19607309 1.00365361 17 1.41801115 1.61807465 1.42158697 1.41872470 0.19040929 1.00264559 18 1.41802525 1.61226923 1.42131256 1.41852784 0.18520944 1.00191512 19 1.41803467 1.60692913 1.42107030 1.41838904 0.18041349 1.00138592 20 1.41804099 1.60199646 1.42085548 1.41829108 0.17597172 1.00100264 21 1.41804523 1.59742274 1.42066420 1.41822188 0.17184260 1.00072515 22 1.41804809 1.59316706 1.42049318 1.41817297 0.16799109 1.00052431 23 1.41805003 1.58919466 1.42033966 1.41813836 0.16438742 1.00037899 24 1.41805134 1.58547582 1.42020132 1.41811387 0.16100611 1.00027389 25 1.41805223 1.58198494 1.42007621 1.41809652 0.15782519 1.00019789 26 1.41805284 1.57869987 1.41996266 1.41808423 0.15482562 1.00014295 27 1.41805326 1.57560132 1.41985928 1.41807551 0.15199081 1.00010326 28 1.41805354 1.57267244 1.41976483 1.41806933 0.14930621 1.00007457 29 1.41805374 1.56989841 1.41967830 1.41806494 0.14675899 1.00005383 30 1.41805387 1.56726614 1.41959880 1.41806182 0.14433784 1.00003886 Table 1: Asymptotic lower/upper bouds for c), R /, ad multipliers for various values of K. 15