{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:OPVOE2LJO5G6TKGRB3MHXLMK45","short_pith_number":"pith:OPVOE2LJ","schema_version":"1.0","canonical_sha256":"73eae26969774de9a8d10ed87bad8ae754a3a01c4b021bdd61951934834b5a21","source":{"kind":"arxiv","id":"1303.5134","version":1},"attestation_state":"computed","paper":{"title":"Bounds on the Number of Huffman and Binary-Ternary Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Angeline Rao, Jian Shen, Yezhou Feng, Ying Liu","submitted_at":"2013-03-21T01:21:41Z","abstract_excerpt":"Huffman coding is a widely used method for lossless data compression because it optimally stores data based on how often the characters occur in Huffman trees. An $n$-ary Huffman tree is a connected, cycle-lacking graph where each vertex can have either $n$ \"children\" vertices connecting to it, or 0 children. Vertices with 0 children are called \\textit{leaves}. We let $h_n(q)$ represent the total number of $n$-ary Huffman trees with $q$ leaves. In this paper, we use a recursive method to generate upper and lower bounds on $h_n(q)$ and get $h_2(q) \\approx (0.1418532)(1.7941471)^q+(0.0612410)(1."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.5134","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2013-03-21T01:21:41Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"c9dda4e580630d98361c7dfa4576e07550d258560199cf8a5a26bef1a29cf257","abstract_canon_sha256":"4daf93b9cc62b9aad524b65b56b719d568927832230ca0a22f4d06844db4f0dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:14.417710Z","signature_b64":"s1B8FDyKKPEIOpkPbta7HWh/q6/vlze8THX8K8ehWizPJE3YiN0PP1ZkJqF+/lRhuw0msTf//CiCgFq62OxaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73eae26969774de9a8d10ed87bad8ae754a3a01c4b021bdd61951934834b5a21","last_reissued_at":"2026-05-18T03:30:14.416894Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:14.416894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds on the Number of Huffman and Binary-Ternary Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Angeline Rao, Jian Shen, Yezhou Feng, Ying Liu","submitted_at":"2013-03-21T01:21:41Z","abstract_excerpt":"Huffman coding is a widely used method for lossless data compression because it optimally stores data based on how often the characters occur in Huffman trees. An $n$-ary Huffman tree is a connected, cycle-lacking graph where each vertex can have either $n$ \"children\" vertices connecting to it, or 0 children. Vertices with 0 children are called \\textit{leaves}. We let $h_n(q)$ represent the total number of $n$-ary Huffman trees with $q$ leaves. In this paper, we use a recursive method to generate upper and lower bounds on $h_n(q)$ and get $h_2(q) \\approx (0.1418532)(1.7941471)^q+(0.0612410)(1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5134","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1303.5134","created_at":"2026-05-18T03:30:14.417029+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.5134v1","created_at":"2026-05-18T03:30:14.417029+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5134","created_at":"2026-05-18T03:30:14.417029+00:00"},{"alias_kind":"pith_short_12","alias_value":"OPVOE2LJO5G6","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"OPVOE2LJO5G6TKGR","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"OPVOE2LJ","created_at":"2026-05-18T12:27:54.935989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45","json":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45.json","graph_json":"https://pith.science/api/pith-number/OPVOE2LJO5G6TKGRB3MHXLMK45/graph.json","events_json":"https://pith.science/api/pith-number/OPVOE2LJO5G6TKGRB3MHXLMK45/events.json","paper":"https://pith.science/paper/OPVOE2LJ"},"agent_actions":{"view_html":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45","download_json":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45.json","view_paper":"https://pith.science/paper/OPVOE2LJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.5134&json=true","fetch_graph":"https://pith.science/api/pith-number/OPVOE2LJO5G6TKGRB3MHXLMK45/graph.json","fetch_events":"https://pith.science/api/pith-number/OPVOE2LJO5G6TKGRB3MHXLMK45/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45/action/storage_attestation","attest_author":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45/action/author_attestation","sign_citation":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45/action/citation_signature","submit_replication":"https://pith.science/pith/OPVOE2LJO5G6TKGRB3MHXLMK45/action/replication_record"}},"created_at":"2026-05-18T03:30:14.417029+00:00","updated_at":"2026-05-18T03:30:14.417029+00:00"}