{"title":"Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves","authors":"M. Bóna, B. Pittel","doi":"10.1002/rsa.21138","DOIUrl":null,"url":null,"abstract":"We prove that for any fixed k$$ k $$ , the probability that a random vertex of a random increasing plane tree is of rank k$$ k $$ , that is, the probability that a random vertex is at distance k$$ k $$ from the leaves, converges to a constant ck$$ {c}_k $$ as the size n$$ n $$ of the tree goes to infinity. We prove that 1−∑j≤kck<22k+3(2k+4)!$$ 1-{\\sum}_{j\\le k}{c}_k<\\frac{2^{2k+3}}{\\left(2k+4\\right)!} $$ , so that the tail of the limiting rank distribution is super‐exponentially narrow. We prove that the latter property holds uniformly for all finite n$$ n $$ as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution {ck}$$ \\left\\{{c}_k\\right\\} $$ . We compute the exact value of ck$$ {c}_k $$ for 0≤k≤3$$ 0\\le k\\le 3 $$ , demonstrating that the limiting expected fraction of vertices with rank ≤3$$ \\le 3 $$ is 0.9997$$ 0.9997 $$ … We show that with probability 1−n−0.99ε$$ 1-{n}^{-0.99\\varepsilon } $$ the highest rank of a vertex in the tree is sandwiched between (1−ε)logn/loglogn$$ \\left(1-\\varepsilon \\right)\\log n/\\mathrm{loglog}n $$ and (1.5+ε)logn/loglogn$$ \\left(1.5+\\varepsilon \\right)\\log n/\\mathrm{loglog}n $$ , and that this rank is asymptotic to logn/loglogn$$ \\log n/\\mathrm{loglog}n $$ with probability 1−o(1)$$ 1-o(1) $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"2013 1","pages":"102 - 129"},"PeriodicalIF":0.9000,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/rsa.21138","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that for any fixed k$$ k $$ , the probability that a random vertex of a random increasing plane tree is of rank k$$ k $$ , that is, the probability that a random vertex is at distance k$$ k $$ from the leaves, converges to a constant ck$$ {c}_k $$ as the size n$$ n $$ of the tree goes to infinity. We prove that 1−∑j≤kck<22k+3(2k+4)!$$ 1-{\sum}_{j\le k}{c}_k<\frac{2^{2k+3}}{\left(2k+4\right)!} $$ , so that the tail of the limiting rank distribution is super‐exponentially narrow. We prove that the latter property holds uniformly for all finite n$$ n $$ as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution {ck}$$ \left\{{c}_k\right\} $$ . We compute the exact value of ck$$ {c}_k $$ for 0≤k≤3$$ 0\le k\le 3 $$ , demonstrating that the limiting expected fraction of vertices with rank ≤3$$ \le 3 $$ is 0.9997$$ 0.9997 $$ … We show that with probability 1−n−0.99ε$$ 1-{n}^{-0.99\varepsilon } $$ the highest rank of a vertex in the tree is sandwiched between (1−ε)logn/loglogn$$ \left(1-\varepsilon \right)\log n/\mathrm{loglog}n $$ and (1.5+ε)logn/loglogn$$ \left(1.5+\varepsilon \right)\log n/\mathrm{loglog}n $$ , and that this rank is asymptotic to logn/loglogn$$ \log n/\mathrm{loglog}n $$ with probability 1−o(1)$$ 1-o(1) $$ .
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.