{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0001","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040b21a46a23e3259b0cd15e87ecdecb88e91562e764c0bf8ff44733cb330cc0","family_id":"qf-tree-property","seed":202610092001,"semantic_template":"qf-tree-property/\"[0,[]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0002","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b33dc97ba6d7785f64f9cdaf32da589e53c13fcba253c7454f62a89e6cf516d","family_id":"qf-tree-property","seed":202610092002,"semantic_template":"qf-tree-property/\"[1,[]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0003","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f2678e70f394c3db46a83faa3d40a9fbfa3ac6fd17ecb85e6365bc6cb0d0a6c","family_id":"qf-tree-property","seed":202610092003,"semantic_template":"qf-tree-property/\"[2,[]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0004","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43e8cdbe949329ae865b7f19fd8d61288823babe178d9222e77f809058299385","family_id":"qf-tree-property","seed":202610092004,"semantic_template":"qf-tree-property/\"[2,[[0,1]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0005","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6206bb7a9cc815e10cd7c0b542a645b564284ffa47d1f429d32f83fc14559109","family_id":"qf-tree-property","seed":202610092005,"semantic_template":"qf-tree-property/\"[3,[]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0006","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dc4151ad6f640af7cbf1c2d7507c2413ae2ad8f9e89ecd07790cd09eee04be9","family_id":"qf-tree-property","seed":202610092006,"semantic_template":"qf-tree-property/\"[3,[[0,1]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0007","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e8285652751cbb73965a811f845f55becc063aa12414aa9588da1db347a4679","family_id":"qf-tree-property","seed":202610092007,"semantic_template":"qf-tree-property/\"[3,[[0,1],[0,2]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1], [1, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0008","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5182bfde89eddf91fda9be300e903bbf7e6f2a312dc6500ff866ac3fc68736cf","family_id":"qf-tree-property","seed":202610092008,"semantic_template":"qf-tree-property/\"[3,[[0,1],[0,2],[1,2]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[0, 1], [1, 2], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0009","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32439e815ef9f4e7b37f0910a09d79224e971e67681e403900990027df23728f","family_id":"qf-tree-property","seed":202610092009,"semantic_template":"qf-tree-property/\"[4,[]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0010","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1295d2cffc94052338a1dfe57291f939e116684543a34d66d256fad84a4ac036","family_id":"qf-tree-property","seed":202610092010,"semantic_template":"qf-tree-property/\"[4,[[0,1]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0011","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef0094ebb0ec95ba6ed5ec48774bf8abded26817759254c126a6fadd2aaf14f","family_id":"qf-tree-property","seed":202610092011,"semantic_template":"qf-tree-property/\"[4,[[0,1],[2,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0012","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01807a2bbc7add8999b12ec3c2767524b31a9f184f8986746308fa800480cdf1","family_id":"qf-tree-property","seed":202610092012,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0013","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84ef507e4b67fa6677d620a710a974f58d1d1334699f9fdcefd80378f7e5a6d0","family_id":"qf-tree-property","seed":202610092013,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0014","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73535ffc4fd045bd04824b1736f0baab48f17a3e9a9ddcac3855264a2472f09e","family_id":"qf-tree-property","seed":202610092014,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0015","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf91091235f9668e777ca00d919514764459d505cc616e835a80a075ccba39e4","family_id":"qf-tree-property","seed":202610092015,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,2]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 2], [2, 3], [3, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0016","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cce6fae322e0856654afa8cef8564b23a9bcc8771eae16fdbecb472314bc3b6","family_id":"qf-tree-property","seed":202610092016,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[1,3],[2,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0017","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd347cd61a3748e06163eb7dfcda6c1afaab79862c09f7ea0157c465b6a3b80e","family_id":"qf-tree-property","seed":202610092017,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 0], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0018","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d274e10c47a8f3957ec36f97161758c0b72432ed72a66991820b01ecce97e3ed","family_id":"qf-tree-property","seed":202610092018,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2],[1,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0019","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6786785aeadbb3030c0628d535100b1c552bfaeec2d9655163ffbeb304f2f55","family_id":"qf-tree-property","seed":202610092019,"semantic_template":"qf-tree-property/\"[4,[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0020","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abf0e45f7211594ec3e8e5f5dfb3a7789c25eca75191669fc17667b51ae45572","family_id":"qf-tree-property","seed":202610092020,"semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[1,3],[2,4]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [1, 2], [2, 3], [3, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0021","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b60818805a7766cd0a3554bc24858a05969bbbf002ecb1ed811c3af3388089bb","family_id":"qf-tree-property","seed":202610092021,"semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[0,3],[0,4]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [0, 2], [0, 3], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0022","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e025d5d5b9b637976c460c3ef163afc3610d1ed7b67f2a6deb2f690a7b064304","family_id":"qf-tree-property","seed":202610092022,"semantic_template":"qf-tree-property/\"[5,[[0,1],[0,2],[1,3],[2,4],[3,4]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0023","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e698159e5bd987df31f9a97f089400ddb99d8084dad5c3b64303c449e1bbf790","family_id":"qf-tree-property","seed":202610092023,"semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[1,3],[2,4],[3,5]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0024","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bc934199ab4e4c83faf616ba5f471c0989811aae16dff33d7450c26baea0986","family_id":"qf-tree-property","seed":202610092024,"semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[0,3],[0,4],[0,5]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"Yes","explanation":"Check all three conditions to obtain Yes. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0025","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fd03afa98d39022cb2f5edae72de949b8c0f19238110bd6205df17a44154786","family_id":"qf-tree-property","seed":202610092025,"semantic_template":"qf-tree-property/\"[6,[[0,1],[0,2],[1,3],[2,4],[3,5],[4,5]]]\""},"fields":{"title":"Nonempty Finite-Graph Tree Eligibility","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. Under the explicit nonempty-tree convention, is the graph nonempty, connected, and acyclic?","answer":{"kind":"short_text","value":"No","explanation":"Check all three conditions to obtain No. The empty graph is not a tree; a singleton without edges is. Edge count alone is insufficient: a triangle plus an isolated vertex has |E|=|V|−1 but is disconnected."},"skills":["Nonempty Finite-Graph Tree Eligibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0026","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79525ad9189b276e7548a3fe16b483c4bb920040a81d5f6af2c4d7512e9a9d07","family_id":"qf-tree-unique-path","seed":202610092026,"semantic_template":"qf-tree-unique-path/{\"E\":[],\"V\":[0],\"s\":0,\"t\":0}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 0 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0]","explanation":"Follow the tree path in traversal order to obtain [0], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0027","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c965d3c16c9dc6c8d09961ad58b6e39fa1740d83da4006845ce20df5726d2a6","family_id":"qf-tree-unique-path","seed":202610092027,"semantic_template":"qf-tree-unique-path/{\"E\":[[0,1]],\"V\":[0,1],\"s\":0,\"t\":1}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 1 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 1]","explanation":"Follow the tree path in traversal order to obtain [0, 1], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0028","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6931fef4f58c12fb2553f606a44d2f9ca06449e9ce3a851ad1965ed5c19d4941","family_id":"qf-tree-unique-path","seed":202610092028,"semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2],\"s\":0,\"t\":2}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 2 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 2]","explanation":"Follow the tree path in traversal order to obtain [0, 2], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0029","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a37a6fa578ea2b68853ad11fa4efd4a37050c36048353861aac90083f7e609a3","family_id":"qf-tree-unique-path","seed":202610092029,"semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"s\":0,\"t\":3}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 3 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 3]","explanation":"Follow the tree path in traversal order to obtain [0, 3], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0030","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f2e776375da761208bf23df82f561701937e6f69bbffd2a6d2436120d540ea","family_id":"qf-tree-unique-path","seed":202610092030,"semantic_template":"qf-tree-unique-path/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"s\":0,\"t\":3}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 3 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 3]","explanation":"Follow the tree path in traversal order to obtain [0, 3], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0031","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8742f3f5dcc45063483a09112218947348efba9e2f6634e93b29d5933ae1628a","family_id":"qf-tree-unique-path","seed":202610092031,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 4 to 4 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[4]","explanation":"Follow the tree path in traversal order to obtain [4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0032","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe62cbb7f9aa07f674cc7b73b26fca49798dd1dc99aee33f1c640a37f5fdbe22","family_id":"qf-tree-unique-path","seed":202610092032,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 4 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 4]","explanation":"Follow the tree path in traversal order to obtain [0, 4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0033","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5c3f81339f3d30470dea2106d8960356bb961fcf0a5ab040b742af331934434","family_id":"qf-tree-unique-path","seed":202610092033,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 4 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 4]","explanation":"Follow the tree path in traversal order to obtain [0, 4], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0034","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb13eabc8690cc0dfaf2bbe653d2872b0b0a7cf040f6ca68e33d5fc098b15f4f","family_id":"qf-tree-unique-path","seed":202610092034,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 5]","explanation":"Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0035","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"169b3945edcf32fdd6322fc885b1475e210442f877df0f901dd04be35db72b4d","family_id":"qf-tree-unique-path","seed":202610092035,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 5]","explanation":"Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0036","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8593a78f0142e66a224965e8c1e5714eee41bb165f27fabff7532bc6c6aaf45","family_id":"qf-tree-unique-path","seed":202610092036,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 5 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[5]","explanation":"Follow the tree path in traversal order to obtain [5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0037","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5556d8b0e0b4e31172e455fbe5e457425bb12e7309ec51b557add9435d147e82","family_id":"qf-tree-unique-path","seed":202610092037,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 5]","explanation":"Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0038","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43011a2a08020c63f023833af639bf3520fdc15ca073cc61ddd75cf7dbade756","family_id":"qf-tree-unique-path","seed":202610092038,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 5]","explanation":"Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0039","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40ef96e9f18cfd62256a087a59f8d5af903e2cecde234fcf3bb1ee52c9c5917","family_id":"qf-tree-unique-path","seed":202610092039,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 5 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 5]","explanation":"Follow the tree path in traversal order to obtain [0, 5], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0040","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d00c765229034b085468972d205c6340e77d242dbec7dff304c0a0fab30951a5","family_id":"qf-tree-unique-path","seed":202610092040,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0041","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf983854cad8021d062f42d3003ec63aa62281045125c7e6b8883e1b7f5eb0b1","family_id":"qf-tree-unique-path","seed":202610092041,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 6 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[6]","explanation":"Follow the tree path in traversal order to obtain [6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0042","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdcd83b09bf9f8a97afeee0908c5568f278c8dc9ca548ff26872b6733e99d81e","family_id":"qf-tree-unique-path","seed":202610092042,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0043","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dea68a0a2fb89a0b2482086eeb352cff251452f8436ec57c7267dad8c60acc31","family_id":"qf-tree-unique-path","seed":202610092043,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0044","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b61d64dd6266dcf029cb4ba6f307c03c6d7a44f8ee968f2b9bc36a298986dccf","family_id":"qf-tree-unique-path","seed":202610092044,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0045","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15dae84cc28e5bdbc761bc3df0aeed5370b41679ff2505fc06ea86108c200c59","family_id":"qf-tree-unique-path","seed":202610092045,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0046","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0965b8398e00ba48d537a332db729e21a0b0506d3ff095916bec7ec9d4678ab7","family_id":"qf-tree-unique-path","seed":202610092046,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 6 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[6]","explanation":"Follow the tree path in traversal order to obtain [6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0047","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9913e212cd026aa253ec7d89f38fc98aa75930d19c393e8bc6d97f2ff757bb95","family_id":"qf-tree-unique-path","seed":202610092047,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0048","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01839b71d9ecd855af5c97480cbdf5b77c6d9f9fe98322a10115fae468b36362","family_id":"qf-tree-unique-path","seed":202610092048,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 5], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0049","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2c82b2bc07a742b020048d880223401b0aa5614fa387f9890c79cc4699e343b","family_id":"qf-tree-unique-path","seed":202610092049,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 6], [3, 5], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0050","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b95c3813dd614025645a9c5f13ed7b182579780a5dee87f105423de0c3cc139a","family_id":"qf-tree-unique-path","seed":202610092050,"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}"},"fields":{"title":"Unique Ordered Path in a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[5, 4], [4, 3], [3, 2], [2, 1], [1, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is confirmed to be a nonempty tree. Give the unique path from 0 to 6 as an ordered vertex list, without repeating vertices.","answer":{"kind":"expression","value":"[0, 6]","explanation":"Follow the tree path in traversal order to obtain [0, 6], not a sorted vertex set. The zero-edge path from a vertex to itself is valid."},"skills":["Unique Ordered Path in a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0051","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05237e1dbf5672c48766bf37732ef4d07eb909045325b3afcc2655335b14fad3","family_id":"qf-tree-leaf-set","seed":202610092051,"semantic_template":"qf-tree-leaf-set/{\"E\":[],\"V\":[0]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0] and E=[]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[]","explanation":"Count normalized incident edges and select degree exactly one, obtaining []. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0052","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97f7ef23b44e34b5c10cd9ed34afb5c47329b9a78605776632c72cbb86bf499c","family_id":"qf-tree-leaf-set","seed":202610092052,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1]],\"V\":[0,1]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1] and E=[[0, 1]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[0, 1]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [0, 1]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0053","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9be823e846f7cc36c9cf86e12f15c3e0fbf1925209e0c75488ca4079704d8809","family_id":"qf-tree-leaf-set","seed":202610092053,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[1, 2]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [1, 2]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0054","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eba832c2721bb6284bb3990200da38a9e977ad2c4ab9fa2c5366c2ad3b2c46e6","family_id":"qf-tree-leaf-set","seed":202610092054,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[1, 2, 3]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0055","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fe7c3340340a5d83215ea6e780676756e2ae3223f3629636c6c9953951fef8a","family_id":"qf-tree-leaf-set","seed":202610092055,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0056","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"697b5ad051a15c799cf4a015272fa109e196e71999ada3d4c16b1f9b63bd3f97","family_id":"qf-tree-leaf-set","seed":202610092056,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3],[0,4]],\"V\":[0,1,2,3,4]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[1, 2, 3, 4]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0057","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d70a0021f55c26dfa27627d1b455d91480e5f2a3fb6ed9da6806340d5ffdc21d","family_id":"qf-tree-leaf-set","seed":202610092057,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[0,4],[1,2]],\"V\":[0,1,2,3,4]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0058","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e16d3349d43b3d2fa7306e87287f5d0c8fb5d1506891f7380d1841ed0a08e5","family_id":"qf-tree-leaf-set","seed":202610092058,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[1,2],[2,3]],\"V\":[0,1,2,3,4]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0059","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dab33f2d3a3e2afdb29ed65322bbb7c3781c2bf6526d865f9b1740e2281991e2","family_id":"qf-tree-leaf-set","seed":202610092059,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0060","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f02b41295fdbc00b51990ba175abc3237c5eab3e7bcc715e9f64dcc606c666","family_id":"qf-tree-leaf-set","seed":202610092060,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0061","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3379102ad2093f3a9d81a411f1e59ed5aed18cf0510f922612d34cec18c65310","family_id":"qf-tree-leaf-set","seed":202610092061,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0062","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7572a94ec4449977d729d2d986ac46fb8680a087fbba46ecc966c0030e2bce7a","family_id":"qf-tree-leaf-set","seed":202610092062,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0063","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84d2984da1030e04aab7b566af8c56c2455513c5d9839e2df1eb143167c77ab","family_id":"qf-tree-leaf-set","seed":202610092063,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0064","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e042af8f8ea3e0c5308d7bffbdbeff87c02fbb7228513e4ab14707f0940610fa","family_id":"qf-tree-leaf-set","seed":202610092064,"semantic_template":"qf-tree-leaf-set/{\"E\":[[0,1],[0,5],[1,2],[2,3],[3,4]],\"V\":[0,1,2,3,4,5]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [0, 5]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[4, 5]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [4, 5]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0065","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7416622cd0fae8647e626fa352c0a16d3bdd2a0be187d3c2abba9827225effc","family_id":"qf-tree-leaf-set","seed":202610092065,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [1, 2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0066","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b42643d8c329de763546767b05d1f1868dff2bc8f14028e05a049a2f6556494","family_id":"qf-tree-leaf-set","seed":202610092066,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [1, 0], [3, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0067","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc1e6eb7d975ca0a506a17360e6b546e8be06c3aacebdda046b7a180d65e0776","family_id":"qf-tree-leaf-set","seed":202610092067,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 1], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0068","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88ec44c71d14d4714b8fd7d9bd44e5e275217dee453e781e4d8944fd285fa5e2","family_id":"qf-tree-leaf-set","seed":202610092068,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0069","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a4a3b4e8481bfa1c52ac6ba64df6df72b7c26bfb3c89d70ef2cded6cc39150","family_id":"qf-tree-leaf-set","seed":202610092069,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 4], [3, 1], [1, 0], [4, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0070","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0827c287af37e89b04d6e2788decbca63679be8f8bf203bb10e0b8628876b9b4","family_id":"qf-tree-leaf-set","seed":202610092070,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [2, 1], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0071","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd7ae0e473b508cc7303ee9841a503ad02cb73b8945d7584bf910a6d4c872307","family_id":"qf-tree-leaf-set","seed":202610092071,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 2], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0072","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f52a866735a859807fbeef1c800b425aed1419924253149c4a191a24ea0c531","family_id":"qf-tree-leaf-set","seed":202610092072,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[4, 3], [3, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[4, 5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [4, 5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0073","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fae632415e0021fed4e7be0b0030d22610c150d4f100a64d950b586467764b3d","family_id":"qf-tree-leaf-set","seed":202610092073,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[3, 5], [4, 2], [2, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[3, 4, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [3, 4, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0074","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f55e49122fb12b90c0588e2e534031e75e4e1953802d3c5ae2b00037479156","family_id":"qf-tree-leaf-set","seed":202610092074,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[2, 6], [3, 5], [4, 1], [1, 0], [5, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[2, 3, 4]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [2, 3, 4]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0075","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb60efea9ed8690f7231c12d521f2996b08d660c5576605d1f6d103429770372","family_id":"qf-tree-leaf-set","seed":202610092075,"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]}"},"fields":{"title":"Degree-One Leaves of a Finite Tree","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4, 5, 6] and E=[[5, 4], [4, 3], [3, 2], [2, 1], [1, 0], [0, 6]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. This is a nonempty tree. List the degree-one leaves in increasing order, using [] when none exist.","answer":{"kind":"expression","value":"[5, 6]","explanation":"Count normalized incident edges and select degree exactly one, obtaining [5, 6]. A singleton vertex has degree zero and is not a leaf under this convention."},"skills":["Degree-One Leaves of a Finite Tree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0076","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be724c2ceb7114c80f7d28ca92984076cc1cf1139e5aa8dbc0a6d158d921bc0c","family_id":"qf-tree-added-edge-cycle","seed":202610092076,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[1,2],[2,3]],\"V\":[0,1,2,3],\"ends\":[1,3]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[0, 1], [1, 2], [2, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 2, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 2, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0077","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95187ee34b0968c35b1fe6d0e2f4797a3e1ba1a233a2b61d168d100fbd4494a7","family_id":"qf-tree-added-edge-cycle","seed":202610092077,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2]],\"V\":[0,1,2],\"ends\":[1,2]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2] and E=[[1, 0], [0, 2]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0078","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d580fd7b546d75fc6e5b953daa6cdafd4c8ff2f7442e1f02322371f90d1c848","family_id":"qf-tree-added-edge-cycle","seed":202610092078,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[1,2]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0079","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0096efa3fd1e9177f5aefb83ebaa41b2851e31b21d546fff76da45431c7e4e9","family_id":"qf-tree-added-edge-cycle","seed":202610092079,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[1,3]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0080","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc2031ba28febeb6c122a9730483b2e96ffd51b0bf9a0ee7c181f30445aad8b","family_id":"qf-tree-added-edge-cycle","seed":202610092080,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,2],[0,3]],\"V\":[0,1,2,3],\"ends\":[2,3]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[1, 0], [2, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 0, 3, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0081","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdb340b41170e8fd3c81fb56853ecd5984461e93a2ef523fdbd873c3b1663359","family_id":"qf-tree-added-edge-cycle","seed":202610092081,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[0,2]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,0 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation":"The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0082","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a8bb9420f678b84fad2e2416eba226ae8778747e7ab43a22ec8f59bd7df3075","family_id":"qf-tree-added-edge-cycle","seed":202610092082,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[1,3]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0083","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e384c418c2f1ebc3fdd46d06d4dba410fc8ca93d2c2b15b18c1b2c14e12eda5d","family_id":"qf-tree-added-edge-cycle","seed":202610092083,"semantic_template":"qf-tree-added-edge-cycle/{\"E\":[[0,1],[0,3],[1,2]],\"V\":[0,1,2,3],\"ends\":[2,3]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3] and E=[[2, 1], [1, 0], [0, 3]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 1, 0, 3, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 1, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0084","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e8c4815ab1da0c466bd42ac274003c4d9911ecf3e715d47516a4a7ed2801600","family_id":"qf-tree-added-edge-cycle","seed":202610092084,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 2, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 2, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0085","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"718f2144eb7a270107e0dbd5bea45e8383f074804f3ab68467e5c6e9975c25ab","family_id":"qf-tree-added-edge-cycle","seed":202610092085,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0086","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41a271e8cced8769ce9e94564e4d6e2a65a2776ef5795a0493c41197a187fd31","family_id":"qf-tree-added-edge-cycle","seed":202610092086,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0087","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5694ecaa677643f0cf7d5c2378ab8c4637b34e2dcbba2eabf8e3657a1d7f10c1","family_id":"qf-tree-added-edge-cycle","seed":202610092087,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 0, 3, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0088","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"162be2ad7c216481dff7291fe11889a6eb28af0dfbd9ebf83cb4b377797c8cf4","family_id":"qf-tree-added-edge-cycle","seed":202610092088,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 0, 4, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0089","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c336f2dbef34dffb84306c3c7197a7a20248c7f308ffaa42d67d42d83d289f73","family_id":"qf-tree-added-edge-cycle","seed":202610092089,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[1, 0], [2, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[3, 0, 4, 3]","explanation":"The genuinely new edge closes the unique original path, giving [3, 0, 4, 3]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0090","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d94064814215f0f4d26f18b5697a8cf93a1b91ce2c0712e9703e452599461a4","family_id":"qf-tree-added-edge-cycle","seed":202610092090,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 0,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation":"The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0091","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22d9577164c422cabbea027ce5e2956585d37880828a25c83ff6ddc1794bf5dd","family_id":"qf-tree-added-edge-cycle","seed":202610092091,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0092","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec84630e9c9b560034107b9a4a3e8fda6a6957c84317e683c0c032179cbd8f0b","family_id":"qf-tree-added-edge-cycle","seed":202610092092,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0093","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf20e393b806dc5a9b54060c3d7dadaab7ab76cff71c555c830c85cff1fa27dc","family_id":"qf-tree-added-edge-cycle","seed":202610092093,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 1, 0, 3, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 1, 0, 3, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0094","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec17ae84e4e626b26e55ed2274a9fff7ccd3930b136d69a09af4d28e9269a819","family_id":"qf-tree-added-edge-cycle","seed":202610092094,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 1, 0, 4, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 1, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0095","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbef03602dcc475e05e8611e4f642afacdd6fdb0ec2b7d6b974cf0111862b082","family_id":"qf-tree-added-edge-cycle","seed":202610092095,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[3, 0, 4, 3]","explanation":"The genuinely new edge closes the unique original path, giving [3, 0, 4, 3]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0096","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1172ef6bf4166eac8e7d5a2561bf1b2825a5246fcd1ac971ecdef094b1bb28a3","family_id":"qf-tree-added-edge-cycle","seed":202610092096,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 0,2 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[0, 1, 2, 0]","explanation":"The genuinely new edge closes the unique original path, giving [0, 1, 2, 0]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0097","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81c39bdbf8ad3ea77ac212c0ed0f2ded2eb75506279152621c3a0a566731b637","family_id":"qf-tree-added-edge-cycle","seed":202610092097,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 3,0 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[0, 1, 2, 3, 0]","explanation":"The genuinely new edge closes the unique original path, giving [0, 1, 2, 3, 0]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0098","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daf661b7b33917013e099c1e2fa9bee182006e68c76e6698e5c7d6b35e0ebfa5","family_id":"qf-tree-added-edge-cycle","seed":202610092098,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 1,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 2, 3, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 2, 3, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0099","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4adfac685c49160cc1a24f6a9b00da7bebecdc7c56252e0a243c75acbeafd13b","family_id":"qf-tree-added-edge-cycle","seed":202610092099,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,1 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[1, 0, 4, 1]","explanation":"The genuinely new edge closes the unique original path, giving [1, 0, 4, 1]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r02-tree-0100","batch_id":"batch-20261007-advance-20261009-r02-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60e17e57bac2dfea6d23976be75a6043928a4da21142785933b603b57f33923e","family_id":"qf-tree-added-edge-cycle","seed":202610092100,"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]}"},"fields":{"title":"Unique Cycle from a Genuinely New Tree Edge","stem":"The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[3, 2], [2, 1], [1, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 2,4 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.","answer":{"kind":"expression","value":"[2, 1, 0, 4, 2]","explanation":"The genuinely new edge closes the unique original path, giving [2, 1, 0, 4, 2]. A repeated existing edge is not an eligible new-edge cycle operation."},"skills":["Unique Cycle from a Genuinely New Tree Edge"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}