{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:2YNE7BIQBZV42G3PBBW3FUYFDX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"41f9c56b8ad87e9001f9dedf2e9a6b0ceb6d66345a05faffd5fa263e5e49fad3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-21T19:38:46Z","title_canon_sha256":"ca9b78f092b82fbc06dea66ddd4744074f48aace40b871d89e5f53e1176da682"},"schema_version":"1.0","source":{"id":"1612.07286","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.07286","created_at":"2026-05-18T00:27:36Z"},{"alias_kind":"arxiv_version","alias_value":"1612.07286v1","created_at":"2026-05-18T00:27:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.07286","created_at":"2026-05-18T00:27:36Z"},{"alias_kind":"pith_short_12","alias_value":"2YNE7BIQBZV4","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"2YNE7BIQBZV42G3P","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"2YNE7BIQ","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:9ad6389f42b09a186a64cec0e95f88fe05f24fb7ffe1ac1fefdb0dc00716d796","target":"graph","created_at":"2026-05-18T00:27:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The register function (or Horton-Strahler number) of a binary tree is a well-known combinatorial parameter. We study a reduction procedure for binary trees which offers a new interpretation for the register function as the maximal number of reductions that can be applied to a given tree. In particular, the precise asymptotic behavior of the number of certain substructures (\"branches\") that occur when reducing a tree repeatedly is determined.\n  In the same manner we introduce a reduction for simple two-dimensional lattice paths from which a complexity measure similar to the register function ca","authors_text":"Benjamin Hackl, Clemens Heuberger, Helmut Prodinger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-21T19:38:46Z","title":"Reductions of Binary Trees and Lattice Paths induced by the Register Function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07286","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:be737074055851b732d1b6450b9467193f837067eabbdf6534da8d7bfb124f92","target":"record","created_at":"2026-05-18T00:27:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"41f9c56b8ad87e9001f9dedf2e9a6b0ceb6d66345a05faffd5fa263e5e49fad3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-21T19:38:46Z","title_canon_sha256":"ca9b78f092b82fbc06dea66ddd4744074f48aace40b871d89e5f53e1176da682"},"schema_version":"1.0","source":{"id":"1612.07286","kind":"arxiv","version":1}},"canonical_sha256":"d61a4f85100e6bcd1b6f086db2d3051dc31e3b390c1e0144292f0c8e72675fda","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d61a4f85100e6bcd1b6f086db2d3051dc31e3b390c1e0144292f0c8e72675fda","first_computed_at":"2026-05-18T00:27:36.420876Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:36.420876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0mbGO2l10reXiFpowPBuGFaHRcssYWQz27NENWTfa/k34Gt8lheW7DIw0/ZM3WgBCSVLIFlPKYKBXjPLAlHjDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:36.421834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.07286","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be737074055851b732d1b6450b9467193f837067eabbdf6534da8d7bfb124f92","sha256:9ad6389f42b09a186a64cec0e95f88fe05f24fb7ffe1ac1fefdb0dc00716d796"],"state_sha256":"b97943d06d006ac498ea6d6b5f8356047d18c298b2f12e288716a72e39e16a02"}