LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES

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1 Acta Mathematica Scientia 2007,27B(3: LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES Su Chun ( Feng Qunqiang ( Liu Jie ( epartment of Statistics and Finance, University of Science and Technology of China, Hefei , China suchun@ustc.edu.cn; fengqq@mail.ustc.edu.cn; jiel@mail.ustc.edu.cn Abstract The authors consider the limiting behavior of various branches in a uniform recursive tree with size growing to infinity. The limiting distribution of ζ n,m, the number of branches with size m in a uniform recursive tree of order n, converges weakly to a Poisson distribution with parameter with convergence of all moments. The size of any large m branch tends to infinity almost surely. Key words Uniform recursive tree, branch, limiting behavior 2000 MR Subject Classification 05C05, 60C05, 60F05 Introduction A tree is a simple connected graph without cycles [6]. The family of recursive tree with size n is a kind of random trees on n particles that attach to each other randomly. The process of generating a recursive tree is as follows (see [2]: let the set of particles be {, 2,, n}, and {p k,i, i =, 2,,k},k =, 2,,, be a sequence of probabilities mass functions, i.e., p k,i 0, k p k,i =, k =, 2,,. i= At step, put all particles in a plane; at step 2, particle 2 attaches to particle ; and at step 3, particle 3 attaches to particle with probability p 2 or to particle 2 with probability p 22. In general, at step k +, particle k + attaches to one of the particles in the set {, 2,, k} with the probabilities p k,i, i =, 2,, k, respectively. After n steps, the resulting tree with the root vertex is called a recursive tree. If p k,i =, i =, 2,, k; k =, 2,,, k that is, at each step, the new particle attaches to a uniformly selected particle from the previous ones independent of previous attachments. We then call it a uniform recursive tree, and denote it by T n. For any natural number k 2, at the k-th (k 2 step we can make k choices, Received April 0, 2005; revised ecember 30, This work was supported by the National Natural Science Foundation of China (06788 and Special Foundation of USTC.

2 56 ACTA MATHEMATICA SCIENTIA Vol.27 Ser.B so (! different recursive trees can be obtained, and each of them occurs with the same probability (!. With many applications, recursive trees have been proposed as models for the spread of epidemics [], the family trees of preserved copies of ancient or medieval texts [2], and pyramid schemes [5], etc. Here we give an example of the model for the spread of epidemics: Example Suppose there exists n persons infected with a specific infectious disease(e.g. SARS in some area, and only one of them is the original case. The second case must have been infected by the original one. Unknowing the law of infection, we suppose that the third case was infected by one of the previous two with the probability /2, each. In general, we suppose the k-th case was infected by one of the previous k cases with respective probabilities k, k = 2, 3,, n. Let vertex k represent the k-th case, and vertex i attaches to vertex j ( i < j n if and only if the jth case was infected by the i-th case. Then we obtain a uniform recursive tree. By this token, such a study of uniform recursive trees can make the law of infection clear to a certain extent. Another kind of random uniform trees is random labeled trees (also called random discrete trees, which for uniform models of randomness amounts to enumerations of various sets of trees, that is, n vertices is connected randomly by undirected edges and each tree occurs with the probability n n+ (see, for example, [0, ]. When the properties of random trees are studied, one usually makes the vertex number n and considers the limiting behaviors (see [8, 0, 5], etc.. From this way, our interest is to study the limiting behavior of the structure of T n. 2 Asymptotic istribution of Branching Number In T n, Let j denote the set of vertices of the jth generation. A subtree with the root in is called a branch, which is also a uniform recursive tree [9]. Branching structure, as an important property of uniform recursive trees, has been studied in the earlier articles [4] and [5] via the combination method. If the size of a branch is m ( m n, we call it an m-branch (m = for a leaf, and let ζ n,m denote the number of the m-branches. In [4], we have shown the exact distribution of ζ n,m. In this section, we shall study its asymptotic distribution based on a method of moments (for this method, see Chern et al. [3], which will improve the relative results in [4]. The factorial moments of a random variable X (with finite moments are the number E X k = E[X(X (X k + ], k 0 (with E X 0 =. The following lemma plays an important role in the method of moments (see Janson et al. [6], for example. Lemma Let X be a random variable with a distribution that is determined by its moments. If X, X 2, are random variables with finite moments such that E X n k E X k for every integer k, then X n X. It is well known that Poisson distribution is determined completely by its moments, and if random variable X has a Poisson distribution with the parameter λ (say X Poi(λ, then

3 No.3 Su et al: LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES 57 the factorial moments of X have the simple form E X k = λ k, k 0. Towards the moments of ζ n,m, we employ a recursive decomposition. The size of the subtree rooted at node k in a uniform recursive tree of order n has been studied by Mahmoud and Smythe [7]. Setting k = 2, one can obtain from their results that the size of the subtree rooted at node 2 is distributed uniformly on {, 2,, }. Here we shall give a simple combinatorial proof of the latter claim via the bijective correspondence between recursive trees and permutations, which was introduced by Stanley [4]. Let A be an indicator of the event A that assumes value if A occurs, and assumes value 0 otherwise. The following lemma is essential for our development. Lemma 2 In uniform recursive trees of order n, for any integer m, ζ n,m =ζn Un, m + {Un=m}, n > m, ( where U n is distributed uniformly on {, 2,, }. Proof Let U n be the size of the subtree rooted at node 2. From the generating of recursive trees, the subtree rooted at 2 must be a branch. Consider it to be special. Then accounts for the contribution from the special branch to the totality ζ n,m if and only if U n = m, and ζ n Un, m accounts for the contribution from outside the special branch. Now, it suffices to show that U n is distributed uniformly on {, 2,, }. As mentioned, Stanley [4] gave the following mapping. Let σ = (σ,, σ be a permutation on {,, n }. Construct a recursive tree with nodes 0,,, by making 0 the root, and defining the parent of node i to be the rightmost element j of σ, both of which precedes i and are less than i. If there is no such element j, then define the parent of i to be the root 0. Finally, to convert to a recursive tree on nodes {, 2,, n}, simply add to each label. For example, the permutation (, 2, 3 corresponds to the linear tree of size 4 where i is the parent of i + for i =, 2, 3; the permutation (3, 2, corresponds to the tree where nodes 2, 3, and 4 are children, each of the root. This mapping is bijective between permutations of {, 2,, } and recursive trees of order n. Note that in this correspondence, the size of the subtree rooted at node 2 is one greater than the number of elements in the corresponding permutation of size that succeed. This number, in turn, is just n minus the position of. The position of is, of course, distributed uniformly on {, 2,, }. The distributional relation ( is useful for the direct computation of (factorial moments. For example, for condition on U n = j, write Eζ n,m = Eζ n Un,m + E {Un=m} = Eζ j,m + P(U n = m j= = Eζ j,m +. j= Note that the boundary conditions ζ j,m 0, for j m; ζ m+,m =Ber( m. iffer a version of the recurrence for Eζ,m from that for Eζ n,m. Observe that because the recurrence ( is

4 58 ACTA MATHEMATICA SCIENTIA Vol.27 Ser.B valid for n > m, we must also take > m to get Eζ n,m = Eζ,m, valid for n > m +. The latter recurrence then unwinds as Eζ n,m = Eζ,m = = Eζ m+,m = m. We thus prove that, Proposition Let ζ n,m be the number of branches of size m in a uniform recursive tree of order n. Then for any integer m, Eζ n,m = m, n > m. Toward the k-th factorial moment calculation for any k 2, we also need the following lemma as a part of the inductive base. Lemma 3 The random variable ζ km+,m has the k-th factorial moment E ζ km+,m k = m k. Proof Note that the random variable ζ n,m is a non negative integer value and cannot exceed m. Then the random variable ζ km+,m cannot exceed k and the range of ζ km+,m is thus {0,,, k}. By the definition of the k-th factorial moment of a random variable, if ζ km+,m assumes one of the values in the set {0,,, k }, then, ζ km+,m k = ζ km+,m (ζ km+,m (ζ km+,m k + = 0, that is, E ζ km+,m k = E [ ] ζ km+,m k {ζkm+,m =k} = k! P(ζ km+,m = k. It suffices to compute the probability P(ζ km+,m = k = /(k! m k. We use induction on k. By the above calculations, it is shown that P(ζ m+,m = = Eζ m+,m = m. Assume now the probability is valid for k, where k 2 is an integer. To have the event {ζ km+,m = k}, we must have an average split: the root node with k branches, each with size m. The special branch will have size m, occurring with probability /(km. The remaining part, which has size (k m +, should consist of the root, and k children of it each, of which fathers m are descendants outside the special branch. By the induction hypothesis, the latter event has probability /[(k!m]. The probability follows. P(ζ km+,m = k = km (k!m = k! m

5 No.3 Su et al: LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES 59 Now we give the main theorem in this section. Theorem Let ζ n,m be the number of branches of size m in a uniform recursive tree of order n. As n, ζ n,m Poi(/m with convergence of all moments. Proof From Lemma, it suffices to prove that for every integer k, E ζ n,m k =, n > mk. (2 mk We use induction on k. As already mentioned, the latter proposition and the lemma show that relation (2 is valid for k = with n > m and for any k with n = km +. Assume now it holds for k (k 2 with n > (k m, where k 2. Now we consider the case k with n > km. It has been done for n = km +. By the recurrence (, we have E ζ n,m k = E ζ n Un,m + {Un=m} k = E ζ n j,m k + E ζ n k,m + k = j,j k j= E ζ j,m k + k E ζ n k,m k. Note that the boundary condition E ζ j,m k 0 for j km, and the fact n k > (k m. By applying the induction hypothesis, the latter recurrence can be rewritten as to E ζ n,m k = j=km+ E ζ j,m k + k (m k. ifferencing a version of the recurrence for from that for n, we simplify the recurrence which implies the relation (2 by Lemma 3. E ζ n,m k = E ζ km+,m k, 3 Limiting Behavior of Large Branches In T n, let ν n denote the size of the biggest branch, i.e., ν n = max{m : ζ n,m }; accordingly, denoted by ν n,k the size of k-th biggest branch: { } max j : ζ n,m k, if this set is nonempty; ν n,k = m=j 0, otherwise. We obtain the following lemma first. Lemma 4 (See [4] In T n, let η n = m= ζ n,m, then η n := m= ζ n,m log n a.s..

6 520 ACTA MATHEMATICA SCIENTIA Vol.27 Ser.B From Lemma 4, it is easy to know that for any fixed k, if n is sufficiently large, there exist more than k branches in the uniform recursive trees of size n almost surely. Therefore, ν n,k, a.s.. We shall prove that for any fixed k, ν n,k, a.s.. Clearly, when a recursive tree with size grows to infinity, ν n,k ν n+,k ν n,k +, thus, ν n,k is nondecreasing in n, and the limit exists. Theorem 2 In uniform recursive trees with size n, for any k N, lim ν n,k =, n Proof Consider the case k = first. Recall that η n denotes the number of all branches, then η n ν n,, from this and Lemma 4, lim inf n log n ν n, n which yields that the theorem holds for k =. For the case k >, note that a.s.. log n lim =, a.s. n η n ( lim ν n,k < = (ν n,k j. n j= i= n=i To prove the theorem, we need only to prove that ( P (ν n,k j = 0, j N. i= n=i For k >, assume now the theorem is established for k. For any 0 < ε <, there exists a sufficiently large natural number i 0 > j +, satisfying i0 exp m < ε. If i > i 0, then By the inductive assumption, m=j+ P(ν i,k j = P(ν i,k j, ν i,k i 0 + P(ν i,k j, ν i,k > i 0 P(ν i,k i 0 + P(ζ i,m = 0, j + m i 0. lim P(ν i,k i 0 = 0, i and from the proof of Theorem 4.6 in [4], we have lim P(ζ i,m = 0, j + m i 0 = exp i i0 m=j+ m < ε, Thus, P ( i= n=i (ν n,k j = lim i P ( (ν n,k j n=i lim i P(ν i,k j < ε.

7 No.3 Su et al: LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES 52 Then the arbitrariness of ε yields that ( P (ν n,k j = 0, j N, i= n=i which implies the theorem is also valid for k, and this finishes the proof. In a rooted tree, the outdegree of a node is the number of its children. If this is finite for each node, we call the tree locally finite. From the above theorem, we get the following subsequence immediately. Corollary In T n, there exists infinitely many branches, which sizes tend to infinity as n almost surely. And then the limit of T n is not locally finite. References Aldous J. The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J iscrete Math, 990, 3: Bergeron F, Flajolet P, Salvy B. Varieties of increasing trees. In: Raoult J C, ed. Proc 7-th Coll Trees in Algebra and Programming (Lecture Notes Comput Sci 58. Berlin: Springer, Chern H, Hwang H, Tsai T. An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms. Journal of Algorithms, 2002, 44: Feng Q, Su C, Hu Z. The structure of branches on uniform recursive trees. Science in China, Ser A, 2005, 35: (In Chinese. 2005, 48: (In English 5 Gastwirth J. A probability model of a pyramid scheme. Amer Statist, 977, 3: Janson S, Luczak T, Rucinski A. Random Graphs. John Wiley & Sons, Mahmoud H, Smythe R T. On the distribution of leaves in rooted subtrees of recursive trees. Ann Appl Prob, 99, : Meir A, Moon J W. Climbing certain types of rooted trees II. Acta Math Acad Sci Hungar, 978, 3( 2: Meir A, Moon J W. On the altitude of nodes in random trees. Canad J Math, 978, 30: Moon J W. Counting Labelled Trees. Canadian Mathematical Congress, 970 Moon J W. The distance between nodes in recursive trees. London Mathematics Society Lecture Notes Series 3. London: Cambridge University Press, Najock, Heyde C. On the number of the terminal vertices in certain random trees with an applications to stemma construction in philology. J Appl Prob, 982, 9: Peterov V V. Limit Theorems of Probability Theory, Sequences of Independent Random Variables. Oxford: Clarendon Press, Stanley R P. Enumerative Combinatorics, Vol I. Monterey, Calif: Wadsworth & Brooks/Cole, Su C, Feng Q, Hu Z. Uniform recursive Trees: Branching structure and simple random downward walk. Journal of Mathematical Analysis And Applications, 2006, 35: Xu J M. Graph Theory and Its Applications (in Chinese. Hefei: USTC Press, 998