scholarly journals Enumeration of Binary Trees and Universal Types

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Generating Functions ◽  
Path Length ◽  
Empirical Distribution ◽  
Applied Mathematics ◽  
Binary Trees ◽  
Wkb Method ◽  
Interesting Problem ◽  
Matched Asymptotics ◽  
Ordered Trees ◽  
International Audience

International audience Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.

scholarly journals Quicksort algorithm again revisited

1999 ◽  
Vol Vol. 3 no. 2 ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Applied Mathematics ◽  
Asymptotic Methods ◽  
Search Tree ◽  
Limiting Distribution ◽  
Wkb Method ◽  
Matched Asymptotics ◽  
Total Path Length ◽  
Double Exponential ◽  
International Audience ◽  
The Right

International audience We consider the standard Quicksort algorithm that sorts n distinct keys with all possible n! orderings of keys being equally likely. Equivalently, we analyze the total path length L(n) in a randomly built \emphbinary search tree. Obtaining the limiting distribution of L(n) is still an outstanding open problem. In this paper, we establish an integral equation for the probability density of the number of comparisons L(n). Then, we investigate the large deviations of L(n). We shall show that the left tail of the limiting distribution is much ''thinner'' (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics.

scholarly journals The height of random binary unlabelled trees

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Nicolas Broutin ◽  
Philippe Flajolet
Keyword(s):  
Large Deviations ◽  
Complex Plane ◽  
Generating Functions ◽  
Binary Trees ◽  
International Audience ◽  
Large Deviations Estimates

International audience This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.

scholarly journals Descendants and ascendants in binary trees

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Binary Tree ◽  
Binary Trees ◽  
Zeilberger’S Algorithm ◽  
International Audience ◽  
Explicit Formulae ◽  
Zeilberger's Algorithm

International audience 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 n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience.

scholarly journals The register function for lattice paths

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Lattice Paths ◽  
Binary Trees ◽  
International Audience

International audience The register function for binary trees is the minimal number of extra registers required to evaluate the tree. This concept is also known as Horton-Strahler numbers. We extend this definition to lattice paths, built from steps $\pm 1$, without positivity restriction. Exact expressions are derived for appropriate generating functions. A procedure is presented how to get asymptotics of all moments, in an almost automatic way; this is based on an earlier paper of the authors.

scholarly journals A repertoire for additive functionals of uniformly distributed m-ary search trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
james Allen fill ◽  
Nevin Kapur
Keyword(s):  
Generating Functions ◽  
Path Length ◽  
Singularity Analysis ◽  
Search Trees ◽  
Hadamard Products ◽  
Space Requirement ◽  
Additive Functionals ◽  
Total Path Length ◽  
International Audience ◽  
Number Of Leaves

International audience Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.

Tight Upper and Lower Bounds on the Path Length of Binary Trees

1994 ◽  
Vol 23 (1) ◽  
pp. 12-23 ◽  
Author(s):  
Alfredo De Santis ◽  
Giuseppe Persiano
Keyword(s):  
Lower Bounds ◽  
Path Length ◽  
Binary Trees ◽  
Upper And Lower Bounds

scholarly journals The b-chromatic number of power graphs

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Brice Effantin ◽  
Hamamache Kheddouci
Keyword(s):  
Chromatic Number ◽  
Binary Trees ◽  
Proper Coloring ◽  
Power Cycles ◽  
International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

Sign in / Sign up

Export Citation Format

Share Document