Alois Panholzer - Academia.edu (original) (raw)
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Papers by Alois Panholzer
A short and computer-free proof (using Euler sums and multiple zeta functions) is provided for a ... more A short and computer-free proof (using Euler sums and multiple zeta functions) is provided for a double sum that was recently computed by Pemantle and Schneider using the computer software Sigma.
Journal of Statistical Planning and Inference, 2002
For the random variable “height of leaf j in a binary tree of size n” we derive closed formulæfor... more For the random variable “height of leaf j in a binary tree of size n” we derive closed formulæfor all moments. The more general case of t-ary trees is also considered.
Discrete Mathematics & Theoretical Computer Science, 1997
There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder ... more There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the internal nodes of a binary tree by the numbers , indicating the sequence in which the nodes are visited. For given (size of the tree) and (a number between 1 and
We give an alternative proof of an identity that appeared recently in Integers .I t is shorter th... more We give an alternative proof of an identity that appeared recently in Integers .I t is shorter than the original one and uses generating functions. In the paper (2) that appeared a few days ago the identity
The Computer Journal, 1998
For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of... more For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of numbers is essential. In the original paper, it was expressed by a recursion. Here, we show how to solve this (and similar) recursions, both explicitly and asymptotically. Some additional information about useful mathematical software is also provided.
Journal of Computational and Applied Mathematics, 2002
We study recursions that are traditional in the context of binary search trees and quicksort with... more We study recursions that are traditional in the context of binary search trees and quicksort with the (non-standard) toll function Hn.
The Electronic Journal of Combinatorics, 1998
The number of descendants of a node in a binary search tree (BST) is the size of the subtree havi... more The number of descendants of a node in a binary search tree (BST) is the size of the subtree having this node as a root; the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach (generating functions and dierential equations) we are able to extend previous results.
In this work we provide a combinatorial analysis of bucket recursive trees, which have been intro... more In this work we provide a combinatorial analysis of bucket recursive trees, which have been introduced previously as a natural generalization of the growth model of recursive trees. Our analysis is based on the description of bucket recursive trees as a special instance of so called b ucket increasing trees, which is a family of combinatorial objects introduced in this
This work is devoted to the analysis of the area under certain lattice paths. The lattice paths o... more This work is devoted to the analysis of the area under certain lattice paths. The lattice paths of interest are associated to a class of 2 2 Polya-Eggenberger urn models with ball replacement matrix M = a 0 c d , with a;d 2 N and c = p a, p 2 N0. We get limiting distributions for the area
We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , ... more We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v1;v2;v3;::: on every path starting at the root of T satisfy v1 < v2 > v3 < v4 > . First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number
We use death processes and embeddings into continuous time in order to analyze several urn models... more We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.
In this work we consider weighted lattice paths in the quarter plane N0 N0. The steps are given b... more In this work we consider weighted lattice paths in the quarter plane N0 N0. The steps are given by (m;n) ! (m 1;n), (m;n) ! (m;n 1) and are weighted as follows: (m;n)! (m 1;n) by m=(m+n) and step (m;n)! (m;n 1) by n=(m+n). The considered lattice paths are absorbed at lines y = x=t s=t with t2 N and
Priority trees are a data structure used for priority queue administration. Un- der the model tha... more Priority trees are a data structure used for priority queue administration. Un- der the model that all permutations of the numbers 1; : : : ; n are equally likely to construct a priority tree of size n we give a detailed average-case analysis of insertion cost measures: we study the recursion depth and the number of key comparisons when
Theoretical Computer Science, 2007
We present a detailed study of left-right-imbalance measures for random binary search trees under... more We present a detailed study of left-right-imbalance measures for random binary search trees under the random permutation model, i.e., where binary search trees are generated by random permutations of f1;2; : : : ; ng. For random binary search trees of size n we study (i) the difierence between the left and the right depth of a randomly chosen node,
The Electronic Journal of Combinatorics, 2006
Simple families of increasing trees can be constructed from simply generated tree families, if on... more Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set f1;:::;ng in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree
This paper deals with the amount of disorder that is left in a permutation after one of its eleme... more This paper deals with the amount of disorder that is left in a permutation after one of its elements has been selected with quickselect with or without median-of-three pivoting. Five measures of disorder are considered: inversions, cycles of length less than or equal to some m, cycles of any length, expected cycle length, and the distance to the identity permutation.
c d , a;d2 N and c2 N0. We obtain limit laws for this class of 2 2 urns by giving estimates for t... more c d , a;d2 N and c2 N0. We obtain limit laws for this class of 2 2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case a = c = d = 1. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, a = d = 1 and c = 0, and corresponding generalizations, a;d2 N and c = 0.
A short and computer-free proof (using Euler sums and multiple zeta functions) is provided for a ... more A short and computer-free proof (using Euler sums and multiple zeta functions) is provided for a double sum that was recently computed by Pemantle and Schneider using the computer software Sigma.
Journal of Statistical Planning and Inference, 2002
For the random variable “height of leaf j in a binary tree of size n” we derive closed formulæfor... more For the random variable “height of leaf j in a binary tree of size n” we derive closed formulæfor all moments. The more general case of t-ary trees is also considered.
Discrete Mathematics & Theoretical Computer Science, 1997
There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder ... more There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the internal nodes of a binary tree by the numbers , indicating the sequence in which the nodes are visited. For given (size of the tree) and (a number between 1 and
We give an alternative proof of an identity that appeared recently in Integers .I t is shorter th... more We give an alternative proof of an identity that appeared recently in Integers .I t is shorter than the original one and uses generating functions. In the paper (2) that appeared a few days ago the identity
The Computer Journal, 1998
For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of... more For the analysis of an algorithm to generate binary trees, the behaviour of a certain sequence of numbers is essential. In the original paper, it was expressed by a recursion. Here, we show how to solve this (and similar) recursions, both explicitly and asymptotically. Some additional information about useful mathematical software is also provided.
Journal of Computational and Applied Mathematics, 2002
We study recursions that are traditional in the context of binary search trees and quicksort with... more We study recursions that are traditional in the context of binary search trees and quicksort with the (non-standard) toll function Hn.
The Electronic Journal of Combinatorics, 1998
The number of descendants of a node in a binary search tree (BST) is the size of the subtree havi... more The number of descendants of a node in a binary search tree (BST) is the size of the subtree having this node as a root; the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach (generating functions and dierential equations) we are able to extend previous results.
In this work we provide a combinatorial analysis of bucket recursive trees, which have been intro... more In this work we provide a combinatorial analysis of bucket recursive trees, which have been introduced previously as a natural generalization of the growth model of recursive trees. Our analysis is based on the description of bucket recursive trees as a special instance of so called b ucket increasing trees, which is a family of combinatorial objects introduced in this
This work is devoted to the analysis of the area under certain lattice paths. The lattice paths o... more This work is devoted to the analysis of the area under certain lattice paths. The lattice paths of interest are associated to a class of 2 2 Polya-Eggenberger urn models with ball replacement matrix M = a 0 c d , with a;d 2 N and c = p a, p 2 N0. We get limiting distributions for the area
We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , ... more We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v1;v2;v3;::: on every path starting at the root of T satisfy v1 < v2 > v3 < v4 > . First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number
We use death processes and embeddings into continuous time in order to analyze several urn models... more We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models.
In this work we consider weighted lattice paths in the quarter plane N0 N0. The steps are given b... more In this work we consider weighted lattice paths in the quarter plane N0 N0. The steps are given by (m;n) ! (m 1;n), (m;n) ! (m;n 1) and are weighted as follows: (m;n)! (m 1;n) by m=(m+n) and step (m;n)! (m;n 1) by n=(m+n). The considered lattice paths are absorbed at lines y = x=t s=t with t2 N and
Priority trees are a data structure used for priority queue administration. Un- der the model tha... more Priority trees are a data structure used for priority queue administration. Un- der the model that all permutations of the numbers 1; : : : ; n are equally likely to construct a priority tree of size n we give a detailed average-case analysis of insertion cost measures: we study the recursion depth and the number of key comparisons when
Theoretical Computer Science, 2007
We present a detailed study of left-right-imbalance measures for random binary search trees under... more We present a detailed study of left-right-imbalance measures for random binary search trees under the random permutation model, i.e., where binary search trees are generated by random permutations of f1;2; : : : ; ng. For random binary search trees of size n we study (i) the difierence between the left and the right depth of a randomly chosen node,
The Electronic Journal of Combinatorics, 2006
Simple families of increasing trees can be constructed from simply generated tree families, if on... more Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set f1;:::;ng in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree
This paper deals with the amount of disorder that is left in a permutation after one of its eleme... more This paper deals with the amount of disorder that is left in a permutation after one of its elements has been selected with quickselect with or without median-of-three pivoting. Five measures of disorder are considered: inversions, cycles of length less than or equal to some m, cycles of any length, expected cycle length, and the distance to the identity permutation.
c d , a;d2 N and c2 N0. We obtain limit laws for this class of 2 2 urns by giving estimates for t... more c d , a;d2 N and c2 N0. We obtain limit laws for this class of 2 2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case a = c = d = 1. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, a = d = 1 and c = 0, and corresponding generalizations, a;d2 N and c = 0.