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THE LEVEL OF NODES IN INCREASING TREES REVISITED ALOIS PANHOLZER AND HELMUT PRODINGER
"... Abstract. Simply generated families of trees are described by the equation T (z) = '(T (z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label 2 f1; : : : ; ng, no label occurs twice, and whenever we proceed from the root to a leaf, the ..."
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Abstract. Simply generated families of trees are described by the equation T (z) = '(T (z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label 2 f1; : : : ; ng, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 di®erent ways. This paper contains results about these families as well as about polynomial families (the function '(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ¸ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j; n!1 such that n¡j is ¯xed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud and others. 1.
THE LEVEL OF NODES IN INCREASING TREES REVISITED ALOIS PANHOLZER AND HELMUT PRODINGER
"... Abstract. Simply generated families of trees are described by the equation T (z) = ϕ(T (z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ {1,..., n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels ar ..."
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Abstract. Simply generated families of trees are described by the equation T (z) = ϕ(T (z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ {1,..., n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j, n → ∞ such that n−j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang’s quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud and others. 1.
1. D. Aldous, The continuum random tree II: an overview, London Math. Soc. Lecture Note
, 2005
"... AMS Subject Classication: 05C05, 05A16, 60F05 Abstract. We consider a recursive procedure for destroying rooted trees and isolating a leaf by removing a random edge and keeping the subtree, which does not contain the original root. For two tree families, the simply generated tree families and increa ..."
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AMS Subject Classication: 05C05, 05A16, 60F05 Abstract. We consider a recursive procedure for destroying rooted trees and isolating a leaf by removing a random edge and keeping the subtree, which does not contain the original root. For two tree families, the simply generated tree families and increasing tree families, we study here the number of random cuts that are necessary to isolate a leaf. We can show limiting distribution results of this parameter for simply generated trees and certain increasing trees.
SPANNING TREE SIZE IN RANDOM BINARY SEARCH TREES
, 2004
"... This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result ( ..."
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Cited by 12 (5 self)
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Quickselect. This parameter, in particular, its first two moments, was studied earlier by Panholzer and Prodinger [Random Structures Algorithms 13 (1998) 189–209]. Here we show also that this normalized parameter has for fixed porder statistics a Gaussian limit law. For p = 1 this gives the wellknown result
unknown title
"... We give an alternative proof of an identity that appeared recently in Integers. It is shorter than the original one and uses generating functions. In the paper [2] that appeared a few days ago the identity Sm: = (x+m+ 1) mX i=0 (¡1)i µ x+ y + i m ¡ i ¶µ y + 2i ..."
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We give an alternative proof of an identity that appeared recently in Integers. It is shorter than the original one and uses generating functions. In the paper [2] that appeared a few days ago the identity Sm: = (x+m+ 1) mX i=0 (¡1)i µ x+ y + i m ¡ i ¶µ y + 2i
Noncrossing trees revisited: cutting down and spanning subtrees
 Discrete Mathematics and Theoretical Computer Science, Proceedings AC, 265–276
, 2003
"... Here we consider two parameters for random noncrossing trees: (i) the number of random cuts to destroy a sizen noncrossing tree and (ii) the spanning subtreesize of p randomly chosen nodes in a sizen noncrossing tree. For both quantities, we are able to characterise for n → ∞ the limiting dis ..."
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Cited by 12 (5 self)
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Here we consider two parameters for random noncrossing trees: (i) the number of random cuts to destroy a sizen noncrossing tree and (ii) the spanning subtreesize of p randomly chosen nodes in a sizen noncrossing tree. For both quantities, we are able to characterise for n → ∞ the limiting distributions. Noncrossing trees are almost conditioned GaltonWatson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter (ii) as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter (i), we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks. Contents
and right length of paths in binary trees or on a question of
 Proceedings AG (2006
"... We consider extended binary trees and study the common right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1,..., n, and the common right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize t ..."
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Cited by 3 (0 self)
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We consider extended binary trees and study the common right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1,..., n, and the common right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf j in a random sizen binary tree when j ∼ ρn with 0 < ρ < 1, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random sizen binary tree.
The degree distribution of thickened trees
 DMTCS Proceedings AI 149–162
, 2008
"... We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global treelike structure while locally looking clustered. This fits with observ ..."
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Cited by 2 (1 self)
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We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global treelike structure while locally looking clustered. This fits with observations obtained from realworld networks. In particular we show that the resulting graphs are scalefree, that is, the degree distribution has an asymptotic power law.
Results 1  10
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