{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R340",
  "content_hash": null,
  "slug": "gtc-artifact-free-tree-generator",
  "type": "artifact",
  "title": "Free tree generator checked against OEIS A000055",
  "summary": "Centroid decomposition enumerates every tree on n vertices exactly once, matching the known counts through n=15.",
  "relevance": "For Graceful tree conjecture, record gtc-artifact-free-tree-generator (“Free tree generator checked against OEIS A000055”) supplies evidence or a replay used to check the packet. The record states: Centroid decomposition enumerates every tree on n vertices exactly once, matching the known counts through n=15.",
  "relevance_source": "recorded",
  "body": "Every tree has one centroid or two adjacent centroids. Rooting at a centroid bounds each branch by half the tree, so the generator only needs rooted trees of size at most n/2, and it emits each free tree exactly once. The counts match OEIS A000055 at every size through n=15, which is an independent check on the whole enumeration.\n\nReusable outside this problem: any exhaustive claim about small trees needs the same generator, and this one is separable from the labeling search.",
  "status": "reproduced",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "counts match A000055 through 15 vertices",
    "bounds": {
      "n": {
        "max": 15
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "program",
    "entrypoint": "free_trees(n)",
    "source": "research/graceful-trees/checks.py",
    "citation": {
      "url": "https://arxiv.org/abs/1403.1564",
      "locator": "research/graceful-trees/checks.py (free_trees)"
    },
    "missing": [
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1403.1564",
    "locator": "research/graceful-trees/checks.py (free_trees)"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R351",
      "title": "Independent exhaustive verification through n=15",
      "object_type": "claim",
      "relation": "enables",
      "direction": "outgoing"
    },
    {
      "slug": "graceful-tree-conjecture",
      "title": "graceful tree conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}