{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:33:43Z","timestamp":1787340823424,"version":"build-2736575974"},"reference-count":8,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[1994,2]]},"abstract":"<jats:p>The external path length of a treeT is the sum of the lengths of the paths from the root to each external node. The maximal path length difference, $\\Delta $, is the difference between the lengths of the longest and shortest such paths<\/jats:p>\n                  <jats:p>Tight lower and upper bounds are proved on the external path length of binary trees with N external nodes and maximal path length difference $\\Delta $ is prescribed.<\/jats:p>\n                  <jats:p>In particular, an upper bound is given that, for each value of $\\Delta $, can be exactly achieved for infinitely many values of N. This improves on the previously known upper bound that could only be achieved up to a factor proportional to N. An elementary proof of the known upper bound is also presented as a preliminary result.<\/jats:p>\n                  <jats:p>Moreover, a lower bound is proved that can be exactly achieved for each value of N and $\\Delta \\leqslant {N \/ 2}$.<\/jats:p>","DOI":"10.1137\/s0097539791197323","type":"journal-article","created":{"date-parts":[[2005,2,24]],"date-time":"2005-02-24T06:39:46Z","timestamp":1109227186000},"page":"12-23","source":"Crossref","is-referenced-by-count":5,"title":["Tight Upper and Lower Bounds on the Path Length of Binary Trees"],"prefix":"10.1137","volume":"23","author":[{"given":"Alfredo","family":"De Santis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giuseppe","family":"Persiano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","unstructured":"H. Cameron, D. Wood,  The Internal Path Length of Red-Black Trees, Research Report, CS-89-50, Computer Science Department, University of Waterloo"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0020822"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"A. De Santis, G. Persiano,  Tight Upper and Lower Bounds on the Path Length of Binary Trees, Tech. report, TR-07-91, Aiken Computation Laboratory, Harvard University,  1991, February","DOI":"10.1007\/BFb0020822"},{"key":"R4","volume-title":"Coding and information theory","author":"Hamming Richard Wesley","year":"1980"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1145\/62044.62048"},{"key":"R6","volume-title":"The art of computer programming. Volume 3","author":"Knuth Donald E.","year":"1973"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1145\/321738.321739"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01180475"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539791197323","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:50:42Z","timestamp":1787338242000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539791197323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,2]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,2]]}},"alternative-id":["10.1137\/S0097539791197323"],"URL":"https:\/\/doi.org\/10.1137\/s0097539791197323","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,2]]}}}