{"id":"https://openalex.org/W2162709814","doi":"https://doi.org/10.37236/1517","title":"A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large","display_name":"A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large","publication_year":2000,"publication_date":"2000-08-01","ids":{"openalex":"https://openalex.org/W2162709814","doi":"https://doi.org/10.37236/1517","mag":"2162709814"},"language":"en","primary_location":{"id":"doi:10.37236/1517","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1517","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r39/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038794095","display_name":"Charles Knessl","orcid":"https://orcid.org/0000-0001-8106-8880"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Charles Knessl","raw_affiliation_strings":["Dept. Mathematics, Statistics & Computer Science Department of Computer Science University of Illinois at Chicago Purdue University Chicago, Illinois 60607-7045 W. Lafayette, IN 47907 U.S.A. U.S.A"],"affiliations":[{"raw_affiliation_string":"Dept. Mathematics, Statistics & Computer Science Department of Computer Science University of Illinois at Chicago Purdue University Chicago, Illinois 60607-7045 W. Lafayette, IN 47907 U.S.A. U.S.A","institution_ids":["https://openalex.org/I39422238"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019536160","display_name":"Wojciech Szpankowski","orcid":"https://orcid.org/0000-0001-9062-0067"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wojciech Szpankowski","raw_affiliation_strings":["Dept. Mathematics, Statistics & Computer Science Department of Computer Science University of Illinois at Chicago Purdue University Chicago, Illinois 60607-7045 W. Lafayette, IN 47907 U.S.A. U.S.A"],"affiliations":[{"raw_affiliation_string":"Dept. Mathematics, Statistics & Computer Science Department of Computer Science University of Illinois at Chicago Purdue University Chicago, Illinois 60607-7045 W. Lafayette, IN 47907 U.S.A. U.S.A","institution_ids":["https://openalex.org/I39422238"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5038794095"],"corresponding_institution_ids":["https://openalex.org/I39422238"],"apc_list":null,"apc_paid":null,"fwci":1.0096,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.76492746,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"7","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8381713628768921},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.7409032583236694},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6920701265335083},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.6568998694419861},{"id":"https://openalex.org/keywords/asymptotic-distribution","display_name":"Asymptotic distribution","score":0.6358767151832581},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.6257237792015076},{"id":"https://openalex.org/keywords/extreme-value-theory","display_name":"Extreme value theory","score":0.5352746844291687},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.4860892593860626},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.4680744409561157},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.4606684446334839},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4288763701915741},{"id":"https://openalex.org/keywords/trie","display_name":"Trie","score":0.41430193185806274},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40792569518089294},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.22508293390274048},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21743324398994446},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15490269660949707},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07643178105354309},{"id":"https://openalex.org/keywords/data-structure","display_name":"Data structure","score":0.0762193500995636}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8381713628768921},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.7409032583236694},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6920701265335083},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.6568998694419861},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.6358767151832581},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.6257237792015076},{"id":"https://openalex.org/C147581598","wikidata":"https://www.wikidata.org/wiki/Q729429","display_name":"Extreme value theory","level":2,"score":0.5352746844291687},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4860892593860626},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4680744409561157},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.4606684446334839},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4288763701915741},{"id":"https://openalex.org/C190290938","wikidata":"https://www.wikidata.org/wiki/Q387015","display_name":"Trie","level":3,"score":0.41430193185806274},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40792569518089294},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.22508293390274048},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21743324398994446},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15490269660949707},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07643178105354309},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.0762193500995636},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1517","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1517","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/1517","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1517","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5219260752","display_name":null,"funder_award_id":"DE-FG","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G6120646659","display_name":"Towards Analytic Information Theory: Data Compression,      Prediction and Universal Coding Through Analytic Methods","funder_award_id":"9804760","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7186324567","display_name":null,"funder_award_id":"DE-FG02-","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8684078510","display_name":null,"funder_award_id":"DE-FG02","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8949489143","display_name":"Data Compression from the String Matching Perspective:      Second Order Properties","funder_award_id":"9415491","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320309036","display_name":"Purdue University","ror":"https://ror.org/02dqehb95"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2162709814.pdf","grobid_xml":"https://content.openalex.org/works/W2162709814.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W938539187","https://openalex.org/W1491940285","https://openalex.org/W1523764855","https://openalex.org/W1549451790","https://openalex.org/W1551073729","https://openalex.org/W1597637931","https://openalex.org/W1965516463","https://openalex.org/W2008188815","https://openalex.org/W2018795275","https://openalex.org/W2023807842","https://openalex.org/W2031314937","https://openalex.org/W2051008728","https://openalex.org/W2088116326","https://openalex.org/W2091851562","https://openalex.org/W2142509031","https://openalex.org/W2160602761","https://openalex.org/W2610179052","https://openalex.org/W2752853835","https://openalex.org/W3198160809"],"related_works":["https://openalex.org/W104884668","https://openalex.org/W3112332783","https://openalex.org/W2120071517","https://openalex.org/W2406489341","https://openalex.org/W2344797854","https://openalex.org/W2113746559","https://openalex.org/W2080508516","https://openalex.org/W180471195","https://openalex.org/W2029271007","https://openalex.org/W3121405494"],"abstract_inverted_index":{"We":[0,28,61,81,103,133,148],"study":[1],"the":[2,6,52,67,87,93,111,120,144],"limiting":[3,88,121],"distribution":[4,69,89,113,122],"of":[5,66,124,154],"height":[7,68,112],"in":[8,12,117],"a":[9,32],"generalized":[10],"trie":[11],"which":[13,118],"external":[14],"nodes":[15],"are":[16,58],"capable":[17],"to":[18,21,74],"store":[19],"up":[20],"$b$":[22,55],"items":[23],"(the":[24],"so":[25],"called":[26],"$b$-tries).":[27],"assume":[29],"that":[30,70,83,105],"such":[31],"tree":[33],"is":[34,90,107,123],"built":[35],"from":[36],"$n$":[37,57],"random":[38],"strings":[39],"(items)":[40],"generated":[41],"by":[42,137],"an":[43,125],"unbiased":[44],"memoryless":[45],"source.":[46],"In":[47],"this":[48,106],"paper,":[49],"we":[50],"discuss":[51],"case":[53,119],"when":[54],"and":[56,143],"both":[59],"large.":[60],"shall":[62],"identify":[63],"five":[64],"regions":[65,76],"should":[71],"be":[72],"compared":[73],"three":[75],"obtained":[77],"for":[78,84,114],"fixed":[79,115],"$b$.":[80],"prove":[82],"most":[85],"$n$,":[86],"concentrated":[91,129],"at":[92],"single":[94],"point":[95,146],"$k_1=\\lfloor":[96],"\\log_2":[97],"(n/b)\\rfloor":[98],"+1$":[99],"as":[100],"$n,b\\to":[101],"\\infty$.":[102],"observe":[104],"quite":[108],"different":[109],"than":[110],"$b$,":[116],"extreme":[126],"value":[127],"type":[128],"around":[130],"$(1+1/b)\\log_2":[131],"n$.":[132],"derive":[134],"our":[135,155],"results":[136],"analytic":[138],"methods,":[139],"namely":[140],"generating":[141],"functions":[142],"saddle":[145],"method.":[147],"also":[149],"present":[150],"some":[151],"numerical":[152],"verification":[153],"results.":[156]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
