A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997-1015.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(of the whole tree) behave independently (i.e. their o springs are iid random variables) Let b n;N denote the number of forests in F (n; N ) weighted according to the probability on F (n; N ) which is the above described probability conditioned on the total progeny. It is well known (see e.g. [15]) that the generating function for those forests is b(z) a(z) a(z) z (a(z) 2) Here a(z) is the generating function for a single tree. Let L n;N (k) be the number of vertices at height k in a random forest in F (n; N ) By [10, Theorem 1.3] we have for n; N 1 such that 2N= n 0 ....

....Proof. The lemma can be proved by an appropriate modi cation of the proof of [6, Theorem 4] When computing the derivatives in (12) only the main term of u k y (z; ua(z) is relevant for the calculation of k (y; b) Set = z) z 00 (a(z) Then, using (z) 1 for z z 0 (see [15] for a detailed expansion) we get by elementary calculations u y (z; u)j u=a(z) y (z; ua(z) j u=a(z) 1 1 y (z; u)j u=a(z) 1 (1 ) 1 (1 ) 1 Plugging these formulas into u j y (z; ua(z) Na and applying Lemma 2 gives (9) 11) This ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30, (1978), 997-1015.

....v t = vg; #A denotes the cardinality of A) conditioned on the number of customers served during the rst busy period (see [ 4] Sec. 7] Another BE application is the number of nodes at some level in a random tree. Consider a simply generated random tree (according to the notion of Meir and Moon [22] or, equivalently, the family tree of a Galton Watson branching process conditioned on the total progeny. Then BE appears as the weak limit of the contour process of this tree, i.e. the process constructed of the distances of the nodes from the root when traversing the tree (for details see ....

A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canadian Journal of Mathematics, 30:997-1015, 1978.

....in the tree between successive external nodes. # n (# ) measures a property of b(# ) that we call the nearly half measure, H n (b(# ) it is the size of the largest subtree with not more than half the external nodes. Though trees have been studied intensively (e.g. 3] 4] 8] 11] and [12]) we are unaware of any previous work on these two features. Theorems 1 and 2 and Lemmas 1 and 2 thus appear to express interesting, new facts about trees, as well as about triangulations. In Section 2 we translate the functions # n and # n into the context of binary trees. We also exploit the ....

A. Meir and J. Moon. On the Altitude of Nodes in Random Trees. Canad. J. Math. 30, 997--1015, 1978.

....trees, in the case of measures with 1 6= 2 the event is concentrated on an exponentially shrinking set of trees, with an exponential rate given by the second term. Examples: The class of Galton Watson trees conditioned on the total size appears in the combinatorial literature, see e.g. [MM78], under the name simply generated trees and is surveyed in [Al91] We look at some interesting examples. Choose the o spring law p( such that p(k) 1 p(0) 1=k: In this case PfjT j = ng 0 if and only if n 1 is divisible by k. The law of T conditional on fjT j = ng is exactly the same as ....

A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997-1015.

....= n 1) n 1) n Note further that the number of unrooted labeled trees of size n equals l n =n since every node in an unrooted tree can be used as a root (and produces n di erent rooted trees) 1.2. Simply Generated Trees. Simply generated trees have been introduced by Meir and Moon [12] and are proper generalizations of several types of rooted trees. Let (x) 0 1 x 2 x be a power series with non negative coecients, in particular we assume that 0 0 and j 0 for some j 2. We then de ne the weight (T ) of a nite rooted tree T by (T ) where D j ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30 (1978), 997-1015.

....of partitional complex constructions in exp log class, like random mappings, 2 regular graphs, children s yards, etc. 15] 4. nodes of given out degree in a random increasing tree in the polynomial variety [2] 5. nodes of given out degree in a random tree in the simply generated family of trees [30, 31]; 6. factorisatio numerorum in (additive) arithmetical semigroups under Axiom A [23] 7. branching compositions of integers introduced in [18, Ch. 8] Example 4. Arithmetical functions. Let f n;k denote the number of factorizations of n into k integer factors greater than 1, n 2, k 1, ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian Journal of Mathematics, 30, 997-1015 (1978).

....of partitional complex constructions in exp log class, like random mappings, 2 regular graphs, children s yards, etc. 15] 4. nodes of given out degree in a random increasing tree in the polynomial variety [2] 5. nodes of given out degree in a random tree in the simply generated family of trees [30, 31]; 6. factorisatio numerorum in (additive) arithmetical semigroups under Axiom A [23] 7. branching compositions of integers introduced in [18, Ch. 8] Example 4. Arithmetical functions. Let f n,k denote the number of factorizations of n into k integer factors greater than 1, n 2, k 1, ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian Journal of Mathematics, 30, 997--1015 (1978).

....2 of the Markov chain setting is replaced here by the additional term 2 (a)I p ( 1 (a) 2 (a) re ecting the large deviations contribution due to the geometry of the tree T . The class of Galton Watson trees conditioned on the total size appears in the combinatorial literature, see e.g. [MM78], under the name simply generated trees and is surveyed in [Al91] We look at some interesting examples. Choose the o spring law p( such that p(k) 1 p(0) 1=k: In this case PfjT j = ng 0 if and only if n 1 is divisible by k. The law of T conditional on fjT j = ng is exactly the same as ....

A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997-1015.

....long runs, surjections, and permutations (Examples 1, 6, and 7) have generating functions with a polar singularity, corresponding to the singular exponent 1. Trees, secondary structures, and noncrossing graphs (Example 2, 3, and 4) which are recursively de ned have singular exponent 2 ; see [24, 49] and Section 8 below. Many properties go along with the conditions of De nition 2. Most notably, the counting sequence associated with a generating function f(z) that is singular systematically obeys an asymptotic law: n 1) This results from the singularity analysis theory ....

....trees with outdegrees in U is the class of trees grafted on a cycle, which are such that their root degree must lie in 1. Let (y) 2 y = The EGF of trees, T , is implicitly de ned by T = z (T ) and one has U = z (T ) It has been rst established by Meir and Moon [49] that the EGF T (z) has systematically a singularity of the square root type (corresponding to failure in the implicit function theorem, see also Lemma 3 below) Precisely, one has T (z) c 1 z= as z , where T is given by Figure 8. A random ternary map( f0; 3g) of size 846 ....

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Meir, A., and Moon, J. W. On the altitude of nodes in random trees. Canadian Journal of Mathematics 30 (1978), 997-1015.

....satisfying the increasing property : labels increase along each branch from the root, itself labelled 0. These are sometimes curiously called recursive trees . The number of such trees with n nodes is well known to equal (n 1) and the trees are in natural correspondence with permutations [8, 11, 39]. Then the number of white balls corresponds to the leaf nodes in the increasing Cayley tree, the statistics of which are known to be given by the Eulerian numbers [8] In fact Mahmoud and Smythe have obtained distributional results about increasing Cayley trees going the other way round and ....

Meir, A., and Moon, J. W. On the altitude of nodes in random trees. Canadian Journal of Mathematics 30 (1978), 997-1015.

....Proof. The lemma can be proved by an appropriate modi cation of the proof of [3, Theorem 4] When computing the derivatives in (12) only the main term of u k y (z; ua(z) is relevant for the calculation of k (y; b) Set = z) z 00 (a(z) Then, using (z) 1 for z z 0 (see [9] for a detailed expansion) we get by elementary calculations u y (z; u)j u=a(z) y (z; ua(z) j u=a(z) 1 1 y (z; u)j u=a(z) 1 (1 ) 1 (1 ) 1 Plugging these formulas into u j y (z; ua(z) Na and applying Lemma 2 gives (9) 11) This ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30, (1978), 997-1015.

.... boustrophedonic search, whose value on any tree of size n is O(n log n) In contrast, the random generation corresponding to the specification ThetaA Gamma A = ThetaA Delta A) has a complexity that behaves like standard path length, which is known to be O(n ) in such varieties of trees [27]. In order to make this discussion precise, we introduce a formal definition. Definition 3 Given two generating functions F and G, F dominates G, in symbols F AE G, if fn gn 1 as n 1: This is only a partial order on generating functions; nonetheless, most naturally occurring generating ....

....= z Delta Phi (T ) Delta GammaT z Delta Phi z Phi(T ) The equation is linear in GammaT . Also, from the defining equation for T , we have 1 Gamma z Phi (T ) T= zT ) Thus, GammaT = zT T z Delta Phi : 25) Asymptotics. We owe to the works of Meir and Moon [27] (see also [35, p. 477] a general analysis of the singularities of the function T . Let be the smallest positive root of the equation Phi( Gamma Phi ( 0: 26) The function T (z) admits a branch point at z = ae, with ae = Phi( 1 : 27) Near this point, we have T (z) ....

Meir, A., and Moon, J. W. On the altitude of nodes in random trees. Canadian Journal of Mathematics 30 (1978), 997--1015. 28

....the proof of this lemma shows that the crucial part is [4, Lemma 3.1] which states that under the assumptions w = O (1) and 1=2 jffj 1 O (jwj) we have for k = O Gamma jwj Gamma1 Delta y k (z; u) Gamma a(z) O Gamma jwff k j Delta : 2. 3) Note that it is well known (see e.g. [16]) that a(z) has a local expansion of the form a(z) Gamma p 2 oe r 1 Gamma z z 0 O fi fi fi fi 1 Gamma z z 0 fi fi fi fi (2.4) around its singulariy z 0 = 1= 0 ( This can e.g. be easily derived by direct application of [7, Theorem 7.1] The assumption i = 1 ensures ....

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian Journal of Mathematics 30 (1978), 997--1015.

....possibilities for that leaf. But this is exactly the symbolic equation for the familiy of C tries, where is used to 3 represent a NIL pointer and 2 pictures a leaf which stores a key. In some sense this concept is related to the notion of simply generated families of trees introduced in [20] and ideas used in [10] The mathematical treatment using generating functions is almost straightforward. We introduce a variable for each type of node. Let v (resp. u, x) mark an internal node of type 0 (resp. 1, 2) and translate the symbolic equation (1) into the corresponding equation for the ....

A. Meir and J. W. Moon, On the Altitude of Nodes in Random Trees, Can. J. Math. 30, 997-1015, 1978

....Section 7. 2. Simply generated trees A simply generated family of trees is defined by a sequence # k , k # 0, of non negative numbers (with # 0 0) each ordered tree T is given a weight # v # d (v) where v ranges over the vertices of T and d(v) is the outdegree (number of children) of v [21]. The corresponding simply generated random tree T n is defined by choosing a tree of order n with probability proportional to its weight (providided that there is any tree of order n with positive weight) It is well known [2] that the simply generated random trees obtained in this way are ....

A. Meir & J.W. Moon, On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997--1015.

....obvious consequences of the closed forms available for Mn ; C k in this particular instance. i) The universal asymptotics of maps. An implicitly de ned function L(z) z (L(z) has in general an isolated singularity of the square root type dictated by a failure of the implicit function theorem [5, 37]: 20) L(z) l 1=2 (1 z= 1=2 O(1 z= l 1=2 0) there the singularity and the singular value are determined by the equations (21) 0 ( 0; The expansion (20) yields in turn the singular expansion of the generating function of maps via M(z) L(z) It ....

....continuable. b) The case = 1=2 covers many generating functions associated to combinatorial structures that are implicitly (or recursively) de ned and have accordingly generating functions with a square root singularity. This includes the varieties of simple trees introduced by Meir and Moon in [37]. Then, one has G(x; 1 2 ) x 2 p exp( x 2 =4) z n ]H k (z) k n n G(x h ; 1 2 ) The law with density proportional to xe x 2 =4 is known as the Rayleigh law: it has been detected in simple trees by Meir and Moon who base their analysis on a Lagrangean change ....

[Article contains additional citation context not shown here]

Meir, A., and Moon, J. W. On the altitude of nodes in random trees. Canadian Journal of Mathematics 30 (1978), 997-1015.

....by virtue of a well known correspondence [28, Chap. 11] Traverse the tree in preorder and output a step of d 1 when a node of outdegree d is encountered. In this way, it is seen that Equation (2) gives the GF of trees counted according to the number of their nodes, an otherwise classical result [31]. By Lagrange inversion, the number of trees comprised of n nodes is Tn = 1 n [w n 1 ] w) n ; where can be directly interpreted as the polynomial of the allowed node (out)degrees. 2. Walks on Z with a infinite set of negative jumps Consider a sequence (e i (k) i a (for a given ....

A. Meir and J. W. Moon. On the altitude of nodes in random trees. anadian Journal of Mathematics, 30:997{ 1015, 1978.

....fl 0 = ae z : z = 1 e 1 Gamma 1 it n and jtj p 2n 1 oe Gamma 0 = ae z : jzj = 1 e and arctan p 2n 1 n Gamma 1 j arg zj oe On fl 0 we substitute z = 1 e Gamma 1 Gamma ff n Delta . Now using the well known expansion for the tree function (see e.g. [19]) on fl 0 a(z) 1 Gamma r 2ff n we obtain the asymptotic relations C r1 C r1 Gamma A r1 Gamma D r1 D r1 Gamma B r1 1 cosh 2 i ae 1 p ff=2 j (17) D r1 Gamma B r1 C r1 Gamma A r1 k1 exp 0 Gammay 1 p 2ff sinh i ae 1 p ff=2 j cosh i ae 1 p ff=2 j 1 A ....

MEIR, A. and MOON, J.W. (1978) On the Altitude of Nodes in Random Trees. Can. J. Math. 30, 997--1015.

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A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997-1015.

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A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30 (1978), 997-1015.

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A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math. 30 (1978), 997-1015.

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A. Meir and J. W. Moon. On the altitude of nodes in random trees. Canadian Journal of Mathematics, 30:997--1015, 1978.

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A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30, (1978), 997-1015.

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A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian Journal of Mathematics 30 (1978), 997-1015.

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A. Meir and J. W. Moon, On the altitude of nodes in random trees, Can. J. Math. 30 (1978), 997-1015.

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