{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-tree",
  "semantic_family_id": "sf-tree",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "非空有限樹：連通、無環與唯一道路"
  },
  "summary": {
    "zh-Hant": "非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。"
  },
  "statement": {
    "zh-Hant": "在有限簡單無向圖G=(V,E)中，本條目約定樹為V非空、任意兩頂點可由道路連接，且沒有簡單環的圖。道路不重複頂點；同一頂點到自身的零邊道路合法。樹中任意兩頂點有唯一道路；葉定義為度數恰1的頂點。"
  },
  "formula": {
    "tex": "T(V,E)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "48a0e94831f2175a2d94ff513d83e488cfa55da07be0da5b3884579bb33ca895"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫V至多10個相異安全整數標籤，E的每項恰兩個V中相异頂點；無向邊方向與重複列項正規化。自環、缺失端點及多重圖語義不在切片。"
    },
    {
      "id": "a2",
      "zh-Hant": "空圖不是樹；單點無邊圖是樹，但其頂點度0，因此本葉慣例下葉集為空。"
    },
    {
      "id": "a3",
      "zh-Hant": "加邊題先要求原圖是樹，u、v在V中相異且原邊不存在。新增這條邊與原唯一道路形成唯一簡單環；已存在邊不適用。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "finite_nonempty_tree_predicate",
      "role_zh": "非空有限樹資格",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "vertex_set",
      "role_zh": "互異頂點集合",
      "scope": "representative_formula"
    },
    {
      "token": "E",
      "role": "unordered_edge_set",
      "role_zh": "簡單無向邊集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限簡單無向圖",
      "reason_zh": "固定無自環、無重邊與無向邊集合語義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "僅有|E|=|V|−1不足以判樹，還需連通；一個三角形加孤點具有這個邊數卻不是樹。空圖與單点的處理明示，不借用模糊的圖形直覺。",
      "唯一道路以實際頂點順序輸出，不列排序後的集合。加新邊的環從兩端較小標籤出發，沿原道路到較大標籤，再以新增邊回起點。葉按度1列遞增集合；一般樹定理仍須形式證明。"
    ],
    "common_misconception_zh": "只看邊數就判樹、將單點度0叫度1葉，或把已有邊重列當成新增環。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限圖切片可完全重算；樹的普遍等價定理及形式Evidence尚未建立。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "固定Mathlib SimpleGraph.IsTree的Connected非空條件，綁定唯一道路、度1葉與非相鄰加邊環。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-tree.md 整理；2026-10-07 脈衝 R02（使用者手動接續 2026-10-09 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}