{"doi":"10.1101/2020.06.16.155184","title":"On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees","abstract":"Abstract Colijn & Plazzotta ( Syst. Biol. 67:113-126, 2018) introduced a scheme for bijectively associating the unlabeled binary rooted trees with the positive integers. First, the rank 1 is associated with the 1-leaf tree. Proceeding recursively, ordered pair ( k 1 , k 2 ), k 1 ⩾ k 2 ⩾ 1, is then associated with the tree whose left subtree has rank k 1 and whose right subtree has rank k 2 . Following dictionary order on ordered pairs, the tree whose left and right subtrees have the ordered pair of ranks ( k 1 , k 2 ) is assigned rank k 1 ( k 1 − 1) / 2 + 1 + k 2 . With this ranking, given a number of leaves n , we determine recursions for a n , the smallest rank assigned to some tree with n leaves, and b n , the largest rank assigned to some tree with n leaves. For n equal to a power of 2, the value of a n is seen to increase exponentially with 2 α n for a constant α ≈ 1.24602; more generally, we show it is bounded a n < 1.5 n . The value of b n is seen to increase with for a constant β ≈ 1.05653. The great difference in the rates of increase for a n and b n indicates that as the index v is incremented, the number of leaves for the tree associated with rank v quickly traverses a wide range of values. We interpret the results in relation to applications in evolutionary biology. Mathematics subject classification 05C05, 92B10, 92D15","journal":"bioRxiv (Cold Spring Harbor Laboratory)","year":2020,"id":130164,"datarank":0.0,"base_score":0.0,"endowment":0.0,"self_citation_contribution":0.0,"citation_network_contribution":0.0,"self_endowment_contribution":0.0,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":0,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.9518,"is_data_producer":false,"deposit_databanks":null,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"2020-01-01","fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":7753,"name":"Noah A. Rosenberg","orcid":"0000-0002-1829-8664","position":0,"is_corresponding":true}],"reference_count":15,"raw_metadata":null,"created_at":"2026-07-18T23:15:53.196774Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}