{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0001","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[]，E=[]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=否；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092001,"parameters":{"V":[],"E":[],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[0,[]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0002","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0]，E=[]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092002,"parameters":{"V":[0],"E":[],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[1,[]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0003","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1]，E=[]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092003,"parameters":{"V":[0,1],"E":[],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[2,[]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0004","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1]，E=[[0,1]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092004,"parameters":{"V":[0,1],"E":[[0,1]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[2,[[0,1]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0005","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2]，E=[]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092005,"parameters":{"V":[0,1,2],"E":[],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[3,[]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0006","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2]，E=[[0,1]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092006,"parameters":{"V":[0,1,2],"E":[[0,1]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[3,[[0,1]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0007","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2]，E=[[0,1],[1,2]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092007,"parameters":{"V":[0,1,2],"E":[[0,1],[1,2]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[3,[[0,1],[0,2]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0008","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2]，E=[[0,1],[1,2],[0,2]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092008,"parameters":{"V":[0,1,2],"E":[[0,1],[1,2],[0,2]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[3,[[0,1],[0,2],[1,2]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0009","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092009,"parameters":{"V":[0,1,2,3],"E":[],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0010","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092010,"parameters":{"V":[0,1,2,3],"E":[[0,1]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0011","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[2,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092011,"parameters":{"V":[0,1,2,3],"E":[[0,1],[2,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[2,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0012","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[1,2]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092012,"parameters":{"V":[0,1,2,3],"E":[[0,1],[1,2]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0013","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[1,2],[2,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092013,"parameters":{"V":[0,1,2,3],"E":[[0,1],[1,2],[2,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0014","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[0,2],[0,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092014,"parameters":{"V":[0,1,2,3],"E":[[0,1],[0,2],[0,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0015","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[1,2],[2,3],[3,1]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=否；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092015,"parameters":{"V":[0,1,2,3],"E":[[1,2],[2,3],[3,1]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,2]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0016","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[1,2],[2,3],[0,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092016,"parameters":{"V":[0,1,2,3],"E":[[0,1],[1,2],[2,3],[0,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,3],[2,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0017","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[1,2],[2,0],[2,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092017,"parameters":{"V":[0,1,2,3],"E":[[0,1],[1,2],[2,0],[2,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0018","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[0,2],[0,3],[1,2],[1,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092018,"parameters":{"V":[0,1,2,3],"E":[[0,1],[0,2],[0,3],[1,2],[1,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2],[1,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0019","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3]，E=[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092019,"parameters":{"V":[0,1,2,3],"E":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0020","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4]，E=[[0,1],[1,2],[2,3],[3,4]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092020,"parameters":{"V":[0,1,2,3,4],"E":[[0,1],[1,2],[2,3],[3,4]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[1,3],[2,4]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0021","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4]，E=[[0,1],[0,2],[0,3],[0,4]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092021,"parameters":{"V":[0,1,2,3,4],"E":[[0,1],[0,2],[0,3],[0,4]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[0,3],[0,4]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0022","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4]，E=[[0,1],[1,2],[2,3],[3,4],[0,4]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092022,"parameters":{"V":[0,1,2,3,4],"E":[[0,1],[1,2],[2,3],[3,4],[0,4]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[1,3],[2,4],[3,4]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0023","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4, 5]，E=[[0,1],[1,2],[2,3],[3,4],[4,5]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092023,"parameters":{"V":[0,1,2,3,4,5],"E":[[0,1],[1,2],[2,3],[3,4],[4,5]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[1,3],[2,4],[3,5]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0024","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4, 5]，E=[[0,1],[0,2],[0,3],[0,4],[0,5]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"是","explanation_zh":"非空=是；連通=是；無環=是。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092024,"parameters":{"V":[0,1,2,3,4,5],"E":[[0,1],[0,2],[0,3],[0,4],[0,5]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[0,3],[0,4],[0,5]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0025","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"非空有限圖的樹資格","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。V=[0, 1, 2, 3, 4, 5]，E=[[0,1],[1,2],[2,3],[3,4],[4,5],[0,5]]，邊無向且重列合併。此圖是否為樹？","answer":{"kind":"short_text","value":"否","explanation_zh":"非空=是；連通=是；無環=否。以DFS分別核非空、連通與無環；空圖不算樹，單點無邊算樹，只看邊數不足以代替資格。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非空有限圖的樹資格"]},"variant":{"family_id":"qf-tree-property","seed":202610092025,"parameters":{"V":[0,1,2,3,4,5],"E":[[0,1],[1,2],[2,3],[3,4],[4,5],[0,5]],"operation":"tree_property","domain":"finite_simple_undirected","semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[1,3],[2,4],[3,5],[4,5]]]\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0026","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0]、E=[]，s=0、t=0。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0]","explanation_zh":"唯一道路=[0]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092026,"parameters":{"V":[0],"E":[],"s":0,"t":0,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[],\"V\":[0],\"s\":0,\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0027","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1]、E=[[0,1]]，s=0、t=1。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"唯一道路=[0, 1]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092027,"parameters":{"V":[0,1],"E":[[0,1]],"s":0,"t":1,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1]],\"V\":[0,1],\"s\":0,\"t\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0028","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2]、E=[[1,0],[0,2]]，s=0、t=2。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"唯一道路=[0, 2]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092028,"parameters":{"V":[0,1,2],"E":[[1,0],[0,2]],"s":0,"t":2,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2],\"s\":0,\"t\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0029","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3]、E=[[1,0],[2,0],[0,3]]，s=0、t=3。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"唯一道路=[0, 3]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092029,"parameters":{"V":[0,1,2,3],"E":[[1,0],[2,0],[0,3]],"s":0,"t":3,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"s\":0,\"t\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0030","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3]、E=[[2,1],[1,0],[0,3]]，s=0、t=3。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"唯一道路=[0, 3]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092030,"parameters":{"V":[0,1,2,3],"E":[[2,1],[1,0],[0,3]],"s":0,"t":3,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"s\":0,\"t\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0031","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，s=4、t=4。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[4]","explanation_zh":"唯一道路=[4]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092031,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"s":4,"t":4,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"s\":4,\"t\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0032","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，s=0、t=4。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"唯一道路=[0, 4]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092032,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"s":0,"t":4,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"s\":0,\"t\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0033","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，s=0、t=4。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"唯一道路=[0, 4]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092033,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"s":0,"t":4,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"s\":0,\"t\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0034","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[1,0],[2,0],[3,0],[4,0],[0,5]]，s=0、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"唯一道路=[0, 5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092034,"parameters":{"V":[0,1,2,3,4,5],"E":[[1,0],[2,0],[3,0],[4,0],[0,5]],"s":0,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"V\":[0,1,2,3,4,5],\"s\":0,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0035","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,1],[1,0],[3,0],[4,0],[0,5]]，s=0、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"唯一道路=[0, 5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092035,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,1],[1,0],[3,0],[4,0],[0,5]],"s":0,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"V\":[0,1,2,3,4,5],\"s\":0,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0036","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,1],[3,1],[1,0],[4,0],[0,5]]，s=5、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[5]","explanation_zh":"唯一道路=[5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092036,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,1],[3,1],[1,0],[4,0],[0,5]],"s":5,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"V\":[0,1,2,3,4,5],\"s\":5,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0037","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[3,2],[2,1],[1,0],[4,0],[0,5]]，s=0、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"唯一道路=[0, 5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092037,"parameters":{"V":[0,1,2,3,4,5],"E":[[3,2],[2,1],[1,0],[4,0],[0,5]],"s":0,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"V\":[0,1,2,3,4,5],\"s\":0,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0038","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,4],[3,1],[1,0],[4,0],[0,5]]，s=0、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"唯一道路=[0, 5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092038,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,4],[3,1],[1,0],[4,0],[0,5]],"s":0,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"V\":[0,1,2,3,4,5],\"s\":0,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0039","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[4,3],[3,2],[2,1],[1,0],[0,5]]，s=0、t=5。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"唯一道路=[0, 5]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092039,"parameters":{"V":[0,1,2,3,4,5],"E":[[4,3],[3,2],[2,1],[1,0],[0,5]],"s":0,"t":5,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[1,2],[2,3],[3,4]],\"V\":[0,1,2,3,4,5],\"s\":0,\"t\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0040","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[1,0],[2,0],[3,0],[4,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092040,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[1,0],[2,0],[3,0],[4,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2],[0,3],[0,4],[0,5],[0,6]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0041","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,1],[1,0],[3,0],[4,0],[5,0],[0,6]]，s=6、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[6]","explanation_zh":"唯一道路=[6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092041,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,1],[1,0],[3,0],[4,0],[5,0],[0,6]],"s":6,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,2]],\"V\":[0,1,2,3,4,5,6],\"s\":6,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0042","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,1],[3,1],[1,0],[4,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092042,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,1],[3,1],[1,0],[4,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,3]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0043","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[2,1],[1,0],[4,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092043,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[2,1],[1,0],[4,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,2],[2,3]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0044","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,4],[3,1],[1,0],[4,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092044,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,4],[3,1],[1,0],[4,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,3],[2,4]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0045","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[2,1],[4,1],[1,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092045,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[2,1],[4,1],[1,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[0,6],[1,2],[1,4],[2,3]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0046","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[4,2],[2,1],[1,0],[5,0],[0,6]]，s=6、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[6]","explanation_zh":"唯一道路=[6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092046,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[4,2],[2,1],[1,0],[5,0],[0,6]],"s":6,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,3],[2,4]],\"V\":[0,1,2,3,4,5,6],\"s\":6,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0047","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[4,3],[3,2],[2,1],[1,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092047,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[4,3],[3,2],[2,1],[1,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,3],[3,4]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0048","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,5],[4,2],[2,1],[1,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092048,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,5],[4,2],[2,1],[1,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,4],[3,5]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0049","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,6],[3,5],[4,1],[1,0],[5,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092049,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,6],[3,5],[4,1],[1,0],[5,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,5],[0,6],[1,4],[2,6],[3,5]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0050","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹兩端的唯一道路","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[5,4],[4,3],[3,2],[2,1],[1,0],[0,6]]，s=0、t=6。列從s到t的頂點次序，不把道路當排序集合。","answer":{"kind":"expression","value":"[0, 6]","explanation_zh":"唯一道路=[0, 6]。在合法樹上用DFS尋找實際次序；兩端相同時零邊道路是單頂點列表，沒有替一般樹定理新增形式證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹兩端的唯一道路"]},"variant":{"family_id":"qf-tree-unique-path","seed":202610092050,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[5,4],[4,3],[3,2],[2,1],[1,0],[0,6]],"s":0,"t":6,"operation":"tree_unique_path","domain":"finite_simple_undirected","semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5]],\"V\":[0,1,2,3,4,5,6],\"s\":0,\"t\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0051","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0]、E=[]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[]","explanation_zh":"度1葉集=[]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092051,"parameters":{"V":[0],"E":[],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[],\"V\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0052","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1]、E=[[0,1]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"度1葉集=[0, 1]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092052,"parameters":{"V":[0,1],"E":[[0,1]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1]],\"V\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0053","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2]、E=[[1,0],[0,2]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[1, 2]","explanation_zh":"度1葉集=[1, 2]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092053,"parameters":{"V":[0,1,2],"E":[[1,0],[0,2]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0054","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3]、E=[[1,0],[2,0],[0,3]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[1, 2, 3]","explanation_zh":"度1葉集=[1, 2, 3]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092054,"parameters":{"V":[0,1,2,3],"E":[[1,0],[2,0],[0,3]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0055","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3]、E=[[2,1],[1,0],[0,3]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3]","explanation_zh":"度1葉集=[2, 3]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092055,"parameters":{"V":[0,1,2,3],"E":[[2,1],[1,0],[0,3]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0056","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[1, 2, 3, 4]","explanation_zh":"度1葉集=[1, 2, 3, 4]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092056,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0057","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4]","explanation_zh":"度1葉集=[2, 3, 4]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092057,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0058","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4]","explanation_zh":"度1葉集=[3, 4]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092058,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0059","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[1,0],[2,0],[3,0],[4,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5]","explanation_zh":"度1葉集=[1, 2, 3, 4, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092059,"parameters":{"V":[0,1,2,3,4,5],"E":[[1,0],[2,0],[3,0],[4,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0060","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,1],[1,0],[3,0],[4,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation_zh":"度1葉集=[2, 3, 4, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092060,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,1],[1,0],[3,0],[4,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0061","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,1],[3,1],[1,0],[4,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation_zh":"度1葉集=[2, 3, 4, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092061,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,1],[3,1],[1,0],[4,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0062","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[3,2],[2,1],[1,0],[4,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4, 5]","explanation_zh":"度1葉集=[3, 4, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092062,"parameters":{"V":[0,1,2,3,4,5],"E":[[3,2],[2,1],[1,0],[4,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0063","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[2,4],[3,1],[1,0],[4,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 5]","explanation_zh":"度1葉集=[2, 3, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092063,"parameters":{"V":[0,1,2,3,4,5],"E":[[2,4],[3,1],[1,0],[4,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0064","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5]、E=[[4,3],[3,2],[2,1],[1,0],[0,5]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[4, 5]","explanation_zh":"度1葉集=[4, 5]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092064,"parameters":{"V":[0,1,2,3,4,5],"E":[[4,3],[3,2],[2,1],[1,0],[0,5]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[1,2],[2,3],[3,4]],\"V\":[0,1,2,3,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0065","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[1,0],[2,0],[3,0],[4,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5, 6]","explanation_zh":"度1葉集=[1, 2, 3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092065,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[1,0],[2,0],[3,0],[4,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3],[0,4],[0,5],[0,6]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0066","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,1],[1,0],[3,0],[4,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4, 5, 6]","explanation_zh":"度1葉集=[2, 3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092066,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,1],[1,0],[3,0],[4,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,2]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0067","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,1],[3,1],[1,0],[4,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4, 5, 6]","explanation_zh":"度1葉集=[2, 3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092067,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,1],[3,1],[1,0],[4,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,3]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0068","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[2,1],[1,0],[4,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation_zh":"度1葉集=[3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092068,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[2,1],[1,0],[4,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,2],[2,3]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0069","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,4],[3,1],[1,0],[4,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 5, 6]","explanation_zh":"度1葉集=[2, 3, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092069,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,4],[3,1],[1,0],[4,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[0,6],[1,3],[2,4]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0070","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[2,1],[4,1],[1,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation_zh":"度1葉集=[3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092070,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[2,1],[4,1],[1,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[0,6],[1,2],[1,4],[2,3]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0071","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,2],[4,2],[2,1],[1,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation_zh":"度1葉集=[3, 4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092071,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,2],[4,2],[2,1],[1,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,3],[2,4]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0072","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[4,3],[3,2],[2,1],[1,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[4, 5, 6]","explanation_zh":"度1葉集=[4, 5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092072,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[4,3],[3,2],[2,1],[1,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,3],[3,4]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0073","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[3,5],[4,2],[2,1],[1,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[3, 4, 6]","explanation_zh":"度1葉集=[3, 4, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092073,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[3,5],[4,2],[2,1],[1,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[0,6],[1,2],[2,4],[3,5]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0074","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[2,6],[3,5],[4,1],[1,0],[5,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[2, 3, 4]","explanation_zh":"度1葉集=[2, 3, 4]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092074,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[2,6],[3,5],[4,1],[1,0],[5,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[0,6],[1,4],[2,6],[3,5]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0075","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"有限樹的度1葉集","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。已給樹 V=[0, 1, 2, 3, 4, 5, 6]、E=[[5,4],[4,3],[3,2],[2,1],[1,0],[0,6]]。葉定義為度1頂點，依遞增次序列其列表。","answer":{"kind":"expression","value":"[5, 6]","explanation_zh":"度1葉集=[5, 6]。由互異無向鄰點數計度；單點的度0不是度1葉，空葉集列[]。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限樹的度1葉集"]},"variant":{"family_id":"qf-tree-leaf-set","seed":202610092075,"parameters":{"V":[0,1,2,3,4,5,6],"E":[[5,4],[4,3],[3,2],[2,1],[1,0],[0,6]],"operation":"tree_leaf_set","domain":"finite_simple_undirected","semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5]],\"V\":[0,1,2,3,4,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0076","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[0,1],[1,2],[2,3]]，加入端點u=3、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 2, 3, 1]","explanation_zh":"新增邊唯一環=[1, 2, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092076,"parameters":{"V":[0,1,2,3],"E":[[0,1],[1,2],[2,3]],"u":3,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[1,2],[2,3]],\"V\":[0,1,2,3],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0077","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2]、E=[[1,0],[0,2]]，加入端點u=2、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation_zh":"新增邊唯一環=[1, 0, 2, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092077,"parameters":{"V":[0,1,2],"E":[[1,0],[0,2]],"u":2,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2],\"ends\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0078","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[1,0],[2,0],[0,3]]，加入端點u=1、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation_zh":"新增邊唯一環=[1, 0, 2, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092078,"parameters":{"V":[0,1,2,3],"E":[[1,0],[2,0],[0,3]],"u":1,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0079","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[1,0],[2,0],[0,3]]，加入端點u=3、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation_zh":"新增邊唯一環=[1, 0, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092079,"parameters":{"V":[0,1,2,3],"E":[[1,0],[2,0],[0,3]],"u":3,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0080","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[1,0],[2,0],[0,3]]，加入端點u=2、v=3的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 0, 3, 2]","explanation_zh":"新增邊唯一環=[2, 0, 3, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092080,"parameters":{"V":[0,1,2,3],"E":[[1,0],[2,0],[0,3]],"u":2,"v":3,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0081","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[2,1],[1,0],[0,3]]，加入端點u=2、v=0的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation_zh":"新增邊唯一環=[0, 1, 2, 0]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092081,"parameters":{"V":[0,1,2,3],"E":[[2,1],[1,0],[0,3]],"u":2,"v":0,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0082","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[2,1],[1,0],[0,3]]，加入端點u=1、v=3的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation_zh":"新增邊唯一環=[1, 0, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092082,"parameters":{"V":[0,1,2,3],"E":[[2,1],[1,0],[0,3]],"u":1,"v":3,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0083","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3]、E=[[2,1],[1,0],[0,3]]，加入端點u=3、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 1, 0, 3, 2]","explanation_zh":"新增邊唯一環=[2, 1, 0, 3, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092083,"parameters":{"V":[0,1,2,3],"E":[[2,1],[1,0],[0,3]],"u":3,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0084","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=1、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation_zh":"新增邊唯一環=[1, 0, 2, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092084,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":1,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0085","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=3、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation_zh":"新增邊唯一環=[1, 0, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092085,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":3,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0086","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=1、v=4的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation_zh":"新增邊唯一環=[1, 0, 4, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092086,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":1,"v":4,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0087","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=3、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 0, 3, 2]","explanation_zh":"新增邊唯一環=[2, 0, 3, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092087,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":3,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0088","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=2、v=4的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 0, 4, 2]","explanation_zh":"新增邊唯一環=[2, 0, 4, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092088,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":2,"v":4,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0089","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[1,0],[2,0],[3,0],[0,4]]，加入端點u=4、v=3的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[3, 0, 4, 3]","explanation_zh":"新增邊唯一環=[3, 0, 4, 3]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092089,"parameters":{"V":[0,1,2,3,4],"E":[[1,0],[2,0],[3,0],[0,4]],"u":4,"v":3,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4],\"ends\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0090","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=0、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation_zh":"新增邊唯一環=[0, 1, 2, 0]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092090,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":0,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0091","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=3、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation_zh":"新增邊唯一環=[1, 0, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092091,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":3,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0092","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=1、v=4的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation_zh":"新增邊唯一環=[1, 0, 4, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092092,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":1,"v":4,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0093","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=3、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 1, 0, 3, 2]","explanation_zh":"新增邊唯一環=[2, 1, 0, 3, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092093,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":3,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0094","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=2、v=4的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 1, 0, 4, 2]","explanation_zh":"新增邊唯一環=[2, 1, 0, 4, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092094,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":2,"v":4,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0095","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[2,1],[1,0],[3,0],[0,4]]，加入端點u=4、v=3的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[3, 0, 4, 3]","explanation_zh":"新增邊唯一環=[3, 0, 4, 3]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092095,"parameters":{"V":[0,1,2,3,4],"E":[[2,1],[1,0],[3,0],[0,4]],"u":4,"v":3,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4],\"ends\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0096","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，加入端點u=0、v=2的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation_zh":"新增邊唯一環=[0, 1, 2, 0]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092096,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"u":0,"v":2,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"ends\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0097","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，加入端點u=3、v=0的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[0, 1, 2, 3, 0]","explanation_zh":"新增邊唯一環=[0, 1, 2, 3, 0]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092097,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"u":3,"v":0,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"ends\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0098","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，加入端點u=1、v=3的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 2, 3, 1]","explanation_zh":"新增邊唯一環=[1, 2, 3, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092098,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"u":1,"v":3,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"ends\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0099","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，加入端點u=4、v=1的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation_zh":"新增邊唯一環=[1, 0, 4, 1]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092099,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"u":4,"v":1,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"ends\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r02-tree-0100","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹加入原本不存在的新邊","stem_zh":"有限簡單無向圖；約定樹須非空、連通且無環。原樹 V=[0, 1, 2, 3, 4]、E=[[3,2],[2,1],[1,0],[0,4]]，加入端點u=2、v=4的邊，原邊不存在。從較小的新增邊端點出發，沿原道路到另一端，再以新邊回起點，列唯一環。","answer":{"kind":"expression","value":"[2, 1, 0, 4, 2]","explanation_zh":"新增邊唯一環=[2, 1, 0, 4, 2]。起點只比較新增邊的兩端，不選全圖或全環的最小標籤；原邊必須不存在，重列已有邊不是此操作。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹加入原本不存在的新邊"]},"variant":{"family_id":"qf-tree-added-edge-cycle","seed":202610092100,"parameters":{"V":[0,1,2,3,4],"E":[[3,2],[2,1],[1,0],[0,4]],"u":2,"v":4,"operation":"tree_added_edge_cycle","domain":"finite_simple_undirected","semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4],\"ends\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}