{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:UXLVVHRATTO42QAI7YQEBAR7T3","short_pith_number":"pith:UXLVVHRA","canonical_record":{"source":{"id":"1705.09688","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-26T19:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"b73a9a175aa4b0343dad5f4e1fa40d6871c2de97a45104cc7fb5f09a6c9686d8","abstract_canon_sha256":"58ac7866112a02c73b2ffbb605d75a92c08b1eaff97cd08271ed6a396d3beaf1"},"schema_version":"1.0"},"canonical_sha256":"a5d75a9e209cddcd4008fe2040823f9eccd63ec4e73cb61de329bddcc6c20686","source":{"kind":"arxiv","id":"1705.09688","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.09688","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"arxiv_version","alias_value":"1705.09688v2","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.09688","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"pith_short_12","alias_value":"UXLVVHRATTO4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"UXLVVHRATTO42QAI","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"UXLVVHRA","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:UXLVVHRATTO42QAI7YQEBAR7T3","target":"record","payload":{"canonical_record":{"source":{"id":"1705.09688","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-26T19:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"b73a9a175aa4b0343dad5f4e1fa40d6871c2de97a45104cc7fb5f09a6c9686d8","abstract_canon_sha256":"58ac7866112a02c73b2ffbb605d75a92c08b1eaff97cd08271ed6a396d3beaf1"},"schema_version":"1.0"},"canonical_sha256":"a5d75a9e209cddcd4008fe2040823f9eccd63ec4e73cb61de329bddcc6c20686","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:10.524841Z","signature_b64":"9XECCvYG0kSBN9O+0x5ffKXWR2Fs64wnTUc1wa2zO3uHHEGDQ9Y/0U+Zjyu1dgTI8mFiZwvGqFbVLBaHEZ5xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5d75a9e209cddcd4008fe2040823f9eccd63ec4e73cb61de329bddcc6c20686","last_reissued_at":"2026-05-18T00:35:10.524470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:10.524470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.09688","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:35:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jyy05IWWiJSgHpyBVTepmKydU0r5pmb6yGR76pApfa+HZ4NIKC2OJAhUZOgJHX7mD0bX+x7aPwFW12xJc9EnDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:27:38.578091Z"},"content_sha256":"6944ddd548fd4e4d52f4c73ad9af7840cae7148d0c4e875a18b6c7ebd7b7974c","schema_version":"1.0","event_id":"sha256:6944ddd548fd4e4d52f4c73ad9af7840cae7148d0c4e875a18b6c7ebd7b7974c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:UXLVVHRATTO42QAI7YQEBAR7T3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Balanced vertices in labeled rooted trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Miklos Bona","submitted_at":"2017-05-26T19:19:55Z","abstract_excerpt":"In a rooted tree, we call a vertex {\\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09688","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:35:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r75tdLAQZRlvUED1/M3oQnMMwfxkqKmhDTUixtFb+6cxBuqJ2ScdBpJKf0EfT3Dk8xOxw7DpIN1V6lid/UUfCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:27:38.578767Z"},"content_sha256":"673299959391d75a28b787224bb21dcb60e9d3baf3a3119e9922efa72f5769c5","schema_version":"1.0","event_id":"sha256:673299959391d75a28b787224bb21dcb60e9d3baf3a3119e9922efa72f5769c5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UXLVVHRATTO42QAI7YQEBAR7T3/bundle.json","state_url":"https://pith.science/pith/UXLVVHRATTO42QAI7YQEBAR7T3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UXLVVHRATTO42QAI7YQEBAR7T3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T18:27:38Z","links":{"resolver":"https://pith.science/pith/UXLVVHRATTO42QAI7YQEBAR7T3","bundle":"https://pith.science/pith/UXLVVHRATTO42QAI7YQEBAR7T3/bundle.json","state":"https://pith.science/pith/UXLVVHRATTO42QAI7YQEBAR7T3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UXLVVHRATTO42QAI7YQEBAR7T3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:UXLVVHRATTO42QAI7YQEBAR7T3","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":"58ac7866112a02c73b2ffbb605d75a92c08b1eaff97cd08271ed6a396d3beaf1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-26T19:19:55Z","title_canon_sha256":"b73a9a175aa4b0343dad5f4e1fa40d6871c2de97a45104cc7fb5f09a6c9686d8"},"schema_version":"1.0","source":{"id":"1705.09688","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.09688","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"arxiv_version","alias_value":"1705.09688v2","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.09688","created_at":"2026-05-18T00:35:10Z"},{"alias_kind":"pith_short_12","alias_value":"UXLVVHRATTO4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"UXLVVHRATTO42QAI","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"UXLVVHRA","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:673299959391d75a28b787224bb21dcb60e9d3baf3a3119e9922efa72f5769c5","target":"graph","created_at":"2026-05-18T00:35:10Z","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":"In a rooted tree, we call a vertex {\\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.","authors_text":"Miklos Bona","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-26T19:19:55Z","title":"Balanced vertices in labeled rooted trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09688","kind":"arxiv","version":2},"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":"sha256:6944ddd548fd4e4d52f4c73ad9af7840cae7148d0c4e875a18b6c7ebd7b7974c","target":"record","created_at":"2026-05-18T00:35:10Z","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":"58ac7866112a02c73b2ffbb605d75a92c08b1eaff97cd08271ed6a396d3beaf1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-26T19:19:55Z","title_canon_sha256":"b73a9a175aa4b0343dad5f4e1fa40d6871c2de97a45104cc7fb5f09a6c9686d8"},"schema_version":"1.0","source":{"id":"1705.09688","kind":"arxiv","version":2}},"canonical_sha256":"a5d75a9e209cddcd4008fe2040823f9eccd63ec4e73cb61de329bddcc6c20686","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5d75a9e209cddcd4008fe2040823f9eccd63ec4e73cb61de329bddcc6c20686","first_computed_at":"2026-05-18T00:35:10.524470Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:10.524470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9XECCvYG0kSBN9O+0x5ffKXWR2Fs64wnTUc1wa2zO3uHHEGDQ9Y/0U+Zjyu1dgTI8mFiZwvGqFbVLBaHEZ5xCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:10.524841Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.09688","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6944ddd548fd4e4d52f4c73ad9af7840cae7148d0c4e875a18b6c7ebd7b7974c","sha256:673299959391d75a28b787224bb21dcb60e9d3baf3a3119e9922efa72f5769c5"],"state_sha256":"a57bbf8cd8956a3ca803d480bfb7373631371ca724c2093dfee8a2c0a00da7ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WfHR+CYJYOa5mOUJSrXAS+32+5Na9/ZnS9R3RDCcbaBcPv2wEwUuzH4KeEPBYFaOAHlvG0s+ozIZyIDJdMAzDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T18:27:38.582538Z","bundle_sha256":"7f6d2f419f72ec2aa9736a0ec5f48afd9f5940058d429069690169992105ea3c"}}