Number of Complete N-ary Subtrees on Galton-Watson Family Trees

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1 Methodol Comput Appl Probab (2006) 8: DOI: /s Number of Complete N-ary Subtrees on Galton-Watson Family Trees George P. Yanev & Ljuben Mutafchiev Received: 5 May 2005 / Revised: 4 January 2006 / Accepted: 18 January 2006 # Springer Science + Business Media, LLC 2006 Abstract We associate with a Bienaymé-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer N, define a complete N-ary tree to be the family tree of a deterministic branching process with offspring generating function s N. We study the random variables V N;n and V N counting the number of disjoint complete N-ary subtrees, rooted at the ancestor, and having height n and 1, respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for PðV N;n > 0Þ and PðV N > 0Þ involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the probability distributions of V N;n and V N. It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws. Keywords Branching process. Family tree. Binary tree. N-ary tree AMS 2000 Subject Classification Primary 60J80. Secondary 05C05 1 Introduction and Main Results Consider the family tree associated with a Bienaymé-Galton-Watson process with the following simple reproduction rules. At generation zero, the process starts with single ancestor called root of the tree. Then each individual in the population has, G. P. Yanev (*) Department of Mathematics, University of South Florida, Tampa, FL 33620, USA gyanev@cas.usf.edu L. Mutafchiev American University in Bulgaria, 2700 Blagoevgrad, Bulgaria ljuben@aubg.bg L. Mutafchiev Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia, Bulgaria

2 224 Methodol Comput Appl Probab (2006) 8: independently of the others, a random number of children distributed according to the offspring distribution with probability generating function (pgf) f ðsþ ¼ 1 p k s k ; satisfying f ð1þ ¼1. Further on we adopt the well-known construction of a family tree generated by a simple branching process where the individuals are the nodes and the parent-child relations define the arcs of the tree in the following manner, see e.g., Harris (1963), Ch.7. Let the ith child of the ancestor be ðiþ and in general ði 1 i 2...i k 1 i k Þ denotes the i k th child of ði 1 i 2...i k 1 Þ. Then, a directed arc is assumed to emanate from ði 1 i 2...i k 1 Þ to ði 1 i 2...i k 1 i k Þ. Since, in our case, the children appear simultaneously, we suppose that the ordering is performed by a chance device independently of the evolution in the process. This scheme produces family trees (also called rooted ordered trees) in which the nodes of height (also known as depth) nðn 0Þ have labels ði 1 i 2...i n Þ, with the ancestor (root) having height 0. The height of a subtree equals the maximum height of its nodes. For fixed integer N 1, define a complete infinite N-ary tree to be the family tree of a deterministic branching process with offspring pgf f ðsþ ¼s N. Further on we will consider rooted subtrees of a family tree. Two such subtrees are called disjoint if they do not have a common node different from the root. These kinds of trees appear, for example, in some computer algorithms; for more details see Knuth (1997). Let fz n :n 1; Z 0 ¼ 1g denote the generation size process, and let T N 1be the height of a complete N-ary subtree rooted in the ancestor; T N ¼ 0ifZ 1 < N. Notice that T 1 is the extinction time of fz n g. The study of the probability N ¼ lim n!1 PðT N > nþ that a Bienaymé-Galton-Watson tree contains an infinite complete N-ary subtree was initiated by Dekking (1991) who considered complete binary (N ¼ 2) subtrees. The general ðn 2Þ case was subsequently investigated in detail by Pakes and Dekking (1991). In particular, they encountered the following phenomenon: if N 2, then there is a critical value m c N for the offspring mean m ¼ f 0 ð1þ such that N ¼ 0ifm < m c N and N > 0ifmm c N. This is qualitatively different from what happens for N ¼ 1 where the probability for non-extinction 1 ¼ 0 if m ¼ m c 1 ¼ 1, except for the trivial case where f ðsþ ¼s. Our work is motivated by the results of Pakes and Dekking (1991). We introduce the random variable V N to be the number of disjoint complete N- ary subtrees with infinite height, rooted at the ancestor of a Bienaymé-Galton- Watson family tree. Clearly N ¼ PðV N > 0Þ. As usual, we assume for the offspring distribution fp k g 1 that p k < 1 for all k and p k > 0 for some k > N. Let N be the set of all positive integers and denote for x; y 0 and any j ¼ 0; 1;... G N ðx; y; jþ ¼ jnþn 1 k¼jn x k k! f ðkþ ðyþ: Pakes and Dekking (1991) showed that PðV N ¼ 0Þ ¼1 N, where 1 N smallest solution in ½0; 1Š of the equation x ¼ G N ð1 x; x;0þ: is the ð1þ

3 Methodol Comput Appl Probab (2006) 8: Our goal is to study the distribution of V N. As the following result shows, the probability mass function (pmf) of V N can be obtained using the Taylor expansion of f ð1þ about the point 1 N. THEOREM 1 If N 2N then for any j ¼ 0; 1; ::: PV ð N ¼ jþ ¼G N ð N ; 1 N ; jþ ð2þ and PðV N ¼ 0Þ ¼1 N is the smallest solution in ½0; 1Š of (1). REMARKS (i) If N ¼ 1, then obviously PðV 1 ¼ 0Þ ¼1 1 ¼ q is the extinction probability of the Galton-Watson process. Now, (2) becomes PðV 1 ¼ jþ ¼ ð1 qþj j! f ðjþ ðqþ; j ¼ 0; 1;...; (ii) which in turn implies that Eðs V 1 Þ¼fðqþð1 qþsþ. This identity follows directly observing that the number of distinct infinite unary trees is equal to the number of first generation nodes having infinite line of descent. 1 Also note that a sufficient condition for PðV N ¼ 0Þ < 1 is given in Pakes and Dekking (1991), Theorem 3. In particular, they show that PðV N ¼0Þ<1ðN 2Þ if 2N jn p j j þ 1 N 1 ð1 p j Þ 2 : The number of complete N-ary subtrees is a measure for the rate of growth (or fertility) of the branching process. In fact, as was pointed out in Dekking (1991), if PðV 2 > 0Þ > 0 then we can say that the branching process grows faster than binary splitting. In the study of the tree structure of branching processes, an important role is played by the process_ total progeny. Denote by n the number of individuals who existed in the first n þ 1 generations, i.e., n ¼ 1 þ Z 1 þ...þ Z n, n ¼ 1; 2;... Obviously, n equals the total number of nodes having height less than or equal to n. Let us also define the random variable V N; n to be the number of disjoint complete N-ary subtrees of height at least n rooted at the ancestor of a Bienaymé-Galton- Watson family tree. Let N;nðsÞ ¼Eðs n ; V N;n > 0Þ and N;n ðsþ ¼Eðs n ; V N;n ¼ 0Þ: ð3þ The following result presents a recursive relation for the joint distribution of V N;n and n. THEOREM 2 If N 2N then for j s j 1 and any j ¼ 0; 1; ::: Es nþ1 ; V N;nþ1 ¼ j ¼ sgn N;n ðsþ; N;n ðsþ; j : ð4þ j¼0 1 The authors are indebted to the referee who pointed out this argument. It implies immediately the result of Theorem 1 for unary trees.

4 226 Methodol Comput Appl Probab (2006) 8: Notice that, if N ¼ 1 and j ¼ 0, then the above recurrence reduces to the wellknown Es nþ1 ð ; Z nþ1 ¼ 0Þ ¼ sf ðeðs n ; Z n ¼ 0ÞÞ, see e.g., Kolchin (1986), p Applications of complete N-ary trees can be found in the analysis of algorithms, see Knuth (1997). Problems of this nature appear also in percolation theory. For instance, Pakes and Dekking (1991) point out a relationship between the model of N-ary complete and infinite subtrees and a construction employed by Chayes et al. (1988) in their study of Mandelbrot_s percolation processes. The existence of N-ary subtrees is also used by Pemantle (1988) in introducing the concept of a N-infinite branching process. Let us also mention potential connections with problems of percolation of binary words on the nodes of locally finite graphs with countably infinite node-sets, see Benjamini and Kesten (1995). We organize our paper as follows. In Section 2 we prove the main results. Sections 3 5 contain some illustrations. In Section 3 we consider the family tree generated by the fractional linear f ðsþ as well as the special case of geometric offspring. In the latter case, V N itself follows a geometric distribution. It turns out that in the Poisson offspring case, given in Section 4, the pmf of V N can be expressed in terms of certain Poisson probabilities. Note that the critical values m c N ðn 2Þ in the Poisson case are less than those in the geometric one. Finally, in Section 5 we consider the one-or-many (i.e., concentrated on two points only) offspring distribution. In this case V N has a pmf given in terms of binomial probabilities. 2 Proofs of the Theorems Proof of Theorem 1: Let us consider PðV N ¼ jþ where j ¼ 1; 2;... Recall that the random variable V N; n equals the number of disjoint complete N-ary subtrees of height n rooted at the ancestor of a Bienaymé-Galton-Watson family tree. First, we will find the pmf of V N; nþ1 using the total probability formula. Indeed, to have j disjoint complete N-ary subtrees rooted at the ancestor node there must be jn þ k ðk 0Þ nodes in the first generation. Each of these nodes can be considered as an ancestor of a family tree rooted at the first generation. Consider the event A N ðlþ ¼ fjn þ l of the Z 1 first generation nodes are ancestors of at least one complete N-ary tree of height ng, where l ¼ 0; 1;...; min fk; N 1g. IfZ 1 ¼ jn þ k then for fixed l the event A N ðlþ has conditional probability PðA N ðlþjz 1 ¼ jn þ kþ ¼ jn þ k ð N;n Þ jnþl ð1 N;n Þ k l jn þ l ð0 l min fk; N 1gÞ; where N; n ¼ 1 PðV N; n ¼ 0Þ and by convention let N; 0 ¼ 1. We have P min fk;n 1g [! A N ðlþjz 1 ¼ jn þ k ¼ min fk;n 1g jn þ k ð N;n Þ jnþl ð1 N;n Þ k l : jn þ l

5 Methodol Comput Appl Probab (2006) 8: Applying the total probability formula and changing the order of summation, we obtain PðV N;nþ1 ¼ jþ ¼ 1 ¼ 1 ¼ N 1 ¼ N 1 p jnþk jnþl N;n ðjn þ lþ! PðZ 1 ¼ jn þ kþp jnþl N;n min fk;n 1g [ A N ðlþjz 1 ¼ jn þ k ( min fk;n 1g ) jn þ k ð N;n Þ jnþl ð1 N;n Þ k l jn þ l 1 k¼l ð jn þ lþ! f ð jnþlþ ð1 N;n Þ ¼ G N ð N;n ; 1 N;n ; jþ p jnþk ð jn þ kþð jn þ k 1Þ:::ðk l þ 1Þð1 N; n Þ k l! By definition N; 0 ¼ 1 and N; n # N as n "1. Letting n!1, we obtain for j 1 PðV N ¼ jþ ¼lim PðV N;nþ1 ¼ jþ ¼G N ð N ; 1 N ; jþ: n!1 Let us now consider the case j ¼ 0. The above recurrence is true for n ¼ 0, i.e., PðV N;1 ¼ 0Þ ¼G N ð1; 0; 0Þ ¼ P N 1 p k: For n 1, using the total probability formula and an argument similar to that for the case j 1, we obtain PðV N;nþ1 ¼ 0Þ ¼ N 1 1 k¼l p k k ð N;n Þ l ð1 N;n Þ k l l ð5þ ¼ N 1 ð N;n Þ l f ðlþ ð1 N;n Þ l! ¼ G N ð N;n ; 1 N;n ;0Þ: Computing the derivative of G N ðx; 1 x;0þ, we get a telescoping sum which after cancelations becomes dg N ðx; 1 x;0þ=dx ¼ð1 xþ N 1 f ðnþ ðxþ=ðn 1Þ! 0 for 0 x 1. Thus, G N ðx; 1 x;0þ is non-decreasing in ½0; 1Š, and therefore 1 N ¼ lim n!1 ð1 N;nþ1 Þ¼lim n!1 PðV N;nþ1 ¼ 0Þ ¼G N ð N ; 1 N ;0Þ is the smallest root in ½0; 1Š of the equation x ¼ G N ð1 x; x;0þ. The proof Í is complete. Clearly (2) implies that P 1 j¼0 PðV N ¼ jþ ¼ P 1 N k f ðkþ ð1 N Þ=k! ¼ f ð1þ ¼1.

6 228 Methodol Comput Appl Probab (2006) 8: Proof of Theorem 2: Let us introduce the notation N;n ðtþ ¼PðV N;n > 0; n ¼ tþ; N;n ðtþ ¼PðV N;n ¼ 0; n ¼ tþ ¼Pð n ¼ tþ N;n ðtþ; where N, n, and t are positive integers. Proceeding as in the proof of Theorem 1, we consider the event A N ðl; tþ ¼A N ðlþ \ f nþ1 ¼ tg; where A N ðlþ is defined in the proof of Theorem 1. For fixed t and l (0 l min ðk; N 1Þ), using the fact that all trees rooted in the first generation grow independently, we compute the conditional probability of A N ðl; tþ given Z 1 ¼ jn þ k to be PðA N ðl; tþjz 1 ¼ jn þ kþ ¼ jn þ k jnþl 0 Y N;n ðn u Þ jn þ l u¼1 jnþk Y v¼jnþlþ1 N;n ðn v Þ; where the summation in P 0 is over all nonnegative integers fn i g jnþk i¼1 such that n i ¼ t 1. Then, the total probability formula implies that P jnþk i¼1 PðV N;nþ1 ¼ j; nþ1 ¼ tþ ¼ 1 ¼ N 1 1 p jnþk k¼l PðZ 1 ¼ jn þ kþ minðk;n 1Þ jn þ k jnþl 0 Y N;n ðn u Þ jn þ l u¼1 PðA N ðl; tþ jz 1 ¼ jn þ kþ jnþk Y v¼jnþlþ1 N;n ðn v Þ: Multiplying both sides of this equality by s t and summing over t, we get Eðs nþ1 ; V N;nþ1 ¼ jþ ¼s N 1 1 ð jn þ lþ! 1 t¼1 0 jnþl Y u¼1 1 k¼l p jnþk ðjn þ kþð jn þ k 1Þ:::ðk l þ 1Þ N;n ðn u Þ Observe that the coefficient of s t 1 in the series can be written as 1 t¼1 0 jnþl Y u¼1 N;n ðn u Þ jnþk Y v¼jnþlþ1 jnþk Y v¼jnþlþ1 N;n ðn v Þs t 1 : N;n ðn v Þs t 1 t 1 h¼0 n 1 þ:::þn jnþl ¼h jnþl Y u¼1 N;n ðn u Þ n jnþlþ1 þ:::þn jnþk ¼t 1 h jnþk Y v¼jnþlþ1 N;n ðn v Þ: The rule of multiplying power series implies that this coefficient equals the coefficient of s t 1 in the power series expansion of " # " 1 jnþl N;n ðiþs # 1 k l i N;n ðiþs i ¼½ N;nðsÞŠ jnþl ½ N;nðsÞŠ k l ; i¼1 i¼1

7 Methodol Comput Appl Probab (2006) 8: where N;n ðsþ and N;n ðsþ are defined in (3). Therefore, Eðs nþ1 ; V N;nþ1 ¼ jþ ¼s N 1 ½ N;n ðsþš jnþl p jnþk ð jn þ kþð jn þ k 1Þ::: ð jn þ lþ! k¼l ðk l þ 1Þ½ N;n ðsþš k l ¼ s N 1 ½ N;n ðsþš jnþl ð jn þ lþ! 1 f ð jnþlþ ð N;n ðsþþ; which coincides with the right-hand side of (4). This completes the proof. 3 Fractional Linear Offspring Í Let f ðsþ be a fractional linear pgf given by f ðsþ ¼1 b 1 p þ bs 1 ps ð6þ and the parameter space fð p; bþ:0< p < 1; 0 < b 1 pg. Then the offspring P distribution is given by the geometric series p k ¼ bp k 1 ; k ¼ 1; 2;...; p 0 ¼ 1 1 k¼1 p k and the offspring mean is m ¼ b=ð1 pþ 2. In the particular case b ¼ pð1 pþ we have p k ¼ð1 pþp k ; k 0 which is the standard geometric distribution with pgf f ðsþ ¼ð1 pþ=ð1 psþ. It can be verified, see Pakes and Dekking (1991), p. 361 if N 2 and Harris (1963), p. 9 if N ¼ 1, that for N 2N 1 pð1 N Þ¼½b=ð1 pþš 1=N ½ p N Š 1 1=N : ð7þ PROPOSITION 1 If the offspring distribution has the fractional linear pgf (6), then V N follows a zero-modified geometric (i.e., fractional linear) distribution given by PðV N ¼ jþ ¼ b pð1 pþ ð1 NÞ j N ð j 1Þ; PðV b N ¼ 0Þ ¼1 pð1 pþ N ð8þ and where EV N ¼ b pð1 pþ N 1 N ; ð9þ N ¼ and N is the largest solution in ½0; 1Š of (7). p N N 1 pð1 N Þ

8 230 Methodol Comput Appl Probab (2006) 8: Proof: Since f ðiþ ðsþ ¼i! bp i 1 =ð1 psþ iþ1 ði 1Þ, we have from (2) for j 1 jnþk N PðV N ¼ jþ ¼ N 1 bð jn þ kþ! p jnþk 1 ð jn þ kþ! ð1 pð1 N ÞÞ jnþkþ1 bp jn 1 jn N 1 N ð p N Þ k ¼ ð1 pð1 N ÞÞ jnþ1 ð1 pð1 N ÞÞ k : Now, setting ð N Þ 1=N ¼ p N =ð1 pð1 N ÞÞ one can obtain the first formula in (8), which in turn leads to (8) and (9). COROLLARY If the offspring distribution is geometric, i.e., p k ¼ð1 pþp k ; k 0, then V N is geometric as well, PðV N ¼ jþ ¼ð1 N Þ j N ð j 0Þ and EV N ¼ N ð1 N Þ 1, where N is the largest solution in ½0; 1Š of ð N þ 1=mÞ N ¼ N 1 N ðn 1Þ. Proof: In the case of geometric offspring (6) holds with b ¼ pð1 pþ and m ¼ p=ð1 pþ. The equation for N follows by inspection from (7). It is also given in Pakes and Dekking (1991), p.361 if N 2. Simple algebraic manipulations show that this equation simplifies to N ¼ N. Now, the rest of the statement follows from (8) and (9). REMARK For geometric offspring with mean m > 1 we have PðV 1 ¼ jþ ¼ð1=mÞð1 1=mÞ j and EV 1 ¼ m 1. In particular, PðV 1 ¼ 0Þ ¼1=m which equals the probability of extinction, see Harris (1963), p. 9. Table 1 lists the probabilities PðV N ¼ jþ, j ¼ 0; 1; 2;...; 9 as well as EV N for 1 N 5. The critical mean values (see Section 1) are as follows: m c 1 ¼ 1, mc 2 ¼ 4, m c 3 ¼ 6:75, mc 4 ¼ 9:481, mc 5 ¼ 12:207. The expected values in the last column provide a measure of how many N-ary subtrees ð1 N 5Þ are supported by the geometric family tree with offspring mean fixed to be m ¼ 13. See also Table 2 below for a comparison with the Poisson offspring case. Í Í 4 Poisson Offspring Consider the case of Poisson offspring distribution with pgf given by f ðsþ ¼e mðs 1Þ ðm > 0Þ: ð10þ Table 1 Probability distribution of V N assuming geometric offspring with m ¼ 13 V N ¼ Q10 EðV N Þ N ¼ N ¼ N ¼ N ¼ N ¼

9 Methodol Comput Appl Probab (2006) 8: Table 2 Probability distribution of V N assuming Poisson offspring with m ¼ 13 V N ¼ Q10 EðV N Þ N ¼ N ¼ N ¼ N ¼ Then, the probability N is the largest solution of ð1 sþe ms ¼ N 1 ðmsþ j =j! (see Pakes and Dekking (1991), p. 364). Since f ðiþ ðsþ ¼m i e mðs 1Þ ði 0Þ, formula (2) becomes PðV N ¼ jþ ¼e m N Therefore, we have the following N 1 j¼0 ðm N Þ jnþk ðjn þ kþ! ; j 0: PROPOSITION 2 If the offspring distribution has the Poisson pgf (10), then PðV N ¼ jþ ¼PðjN Y N jn þ N 1Þ; where Y N has the Poisson pmf PðY N ¼ kþ ¼ðm N Þ k e mn =k! k ¼ 0; 1; 2;... and N is the largest solution in ½0; 1Š of equation (11). Notice that V 1 has a Poisson distribution with parameter m 1. To calculate the critical value m c N that yields a non-zero solution N c in ½0; 1Š of equation (11) we first notice that the product y ¼ m c N N c satisfies the equations y N =ðn 1Þ! þ N 1 y j =j! ¼ e y ; see Pakes and Dekking (1991), p Following their way of calculation, one can find m c N and N c by substituting the solution of (12) into my N 1 =ðn 1Þ! ¼ e y : ð13þ In P case of binary trees, one can also use the Cayley_s tree function yðzþ ¼ 1 k¼1 kk 1 z k =k! (see e.g., Odlyzko (1995), Section 6.2) evaluated at z ¼ 1=m c N for the solution of (12). Inserting it into (12), we obtain m c 2 ¼ 3:3509 and 2 c ¼ 0:5352. Our final remark concerns the case m!1. It is easily seen that Proposition 2 and the normal approximation of the Poisson distribution imply a local limit theorem for V N. Moreover, Pakes and Dekking (1991) showed that in this case N! 1. This enables one to centralize and scale the limiting variable V N in terms of the single parameter m only. j¼0 ð11þ ð12þ

10 232 Methodol Comput Appl Probab (2006) 8: Table 2 gives the probabilities PðV N ¼ jþ, j ¼ 0; 1; 2;...; 9 as well as EV N for 2 N 5. The critical mean values are as follows: m c 2 ¼ 3:3509, mc 3 ¼ 5:1494, m c 4 ¼ 6:7993, mc 5 ¼ 8: One-or-many Offspring In this section we consider a two-parameter family of 1-or-r offspring distributions defined for some p 2ð0; 1Þ by p 1 ¼ 1 p and p r ¼ p, where r > N > 1. Its pgf is f ðsþ ¼ð1 pþs þ ps r and thus f 0 ðsþ ¼1 p þ prs r 1 and f ðkþ ðsþ ¼prðr 1Þ... ðr k þ 1Þs r k ð2 k rþ. The probability N is the largest solution in ½0; 1Š of s ¼ p r k¼n r s k ð1 sþ r k k (see again Pakes and Dekking (1991), p.366). Applying (2) it is not difficult to obtain PðV N ¼ 0Þ ¼1 p þ p N 1 r N k k ð1 NÞ r k and for j ¼ 1; 2;... and r jn jnþu PðV N ¼ jþ ¼p r N k k ð1 NÞ r k ; k¼jn where U ¼minfN 1; r jng. Let B r ð N Þ denote a binomialðr; N Þrandom variable. PROPOSITION 3 If the offspring pgf is f ðsþ ¼ð1 pþs þ ps r ð1 N < rþ and N is the largest solution in ½0; 1Š of (14), then PðV N ¼ 0Þ ¼1 p þ ppðb r ð N ÞN 1Þ and for j ¼ 1; 2;... ð14þ PðV N ¼ jþ ¼pPð jn B r ð N ÞjN þ UÞ if jn r; ð15þ where U ¼ minfn 1; r jng and PðV N ¼ jþ ¼0ifjN > r. The expected value of V N is where ½xŠ is the integer part of x. EV N ¼ p ½r=NŠ jpð jn B r ð N ÞjN þ UÞ; j¼1 Table 3 Probability distribution of V N assuming 1-or-14 offspring with p ¼ 0:93 (m ¼ 13:09) V N ¼ EðV N Þ N ¼ N ¼ N ¼ N ¼

11 Methodol Comput Appl Probab (2006) 8: In particular, if r ¼ N þ 1orr¼Nþ2 and N > 2, then (15) implies that V N takes on values 0 or 1; if N ¼ 2 and r ¼ 4, then V N takes on values 0, 1, or 2. Table 3 provides some numerical illustrations. Note that the offspring mean m ¼ 13:09 enables comparisons with Tables 1 and 2. It is interesting to point out the following relationship between the 1-or-r and Poisson offspring cases. There exists (see Pakes and Dekking (1991)) a critical value p c N such that for p ¼ pc N equation (14) has a single solution N c in ð0; 1Þ. Suppose that lim r!1 ðrn c Þ!y, where y satisfies (13) and (12). Then, applying Theorem 7, Pakes and Dekking (1991), one can obtain that V N ðrþ converges in distribution to V N ðyþ, where V N ðrþ and V N ðyþ are copies of V N assuming one-or-many and Poisson offspring with mean m c N, respectively. Acknowledgments We thank the referee for his valuable comments and suggestions and especially for his help to eliminate some defects in Proposition 1. This work was done during L. Mutafchiev_s visitat the Mathematics Department of the University of South Florida in academic year. He thanks for the hospitality and support. G. Yanev is partially supported by NFSI-Bulgaria, MM-1101/2001. References I. Benjamini, and H. Kesten, BPercolation of arbitrary words in f0; 1g N, Annals of Probability vol. 23 pp , J. L. Chayes, L. Chayes, and R. Durret, BConnectivity properties of Mandelbrot_s percolation process, Probability Theory and Related Fields vol. 77 pp , F. M. Dekking, BBranching processes that grow faster than binary splitting, American Mathematical Monthly vol. 98 pp , T. E. Harris, The Theory of Branching Processes, Springer, Berlin, D. E. Knuth, The Art of Computer Programming, vol. 1: Fundamental Algorithms, 3rd ed., Addison- Wesley: Reading, Mass, V. F. Kolchin, Random Mappings, Optimization Software, Inc.: New York, A. M. Odlyzko, BAsymptotic enumeration methods. In R. Graham, M. Grötshel and L. Lovász (ed.), Handbook of Combinatorics, vol.2, pp , Elsevier Sci., A. G. Pakes, and F. M. Dekking, BOn family trees and subtrees of simple branching processes, Journal of Theoretical Probability vol. 4 pp , R. Pemantle, BPhase transition in reinforced random walk and RWRE on trees, Annals of Probability vol. 16 pp , 1988.