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Little is known about this question apart from a few simple observations. We show that the number of trees on $n$ vertices which can be distinguished by their associated integer sequences is $e^{\\Omega((\\log n)^{3/2})}$. The proof strategy involves constructing a large collection of caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott prob"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1109.0522","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-02T19:08:27Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fcfcb734fe0558312287bd03986236d32444fd08450d131e7a5c7b8eeedccea6","abstract_canon_sha256":"8c2cce0b746d0ea908c34db8ea9b2b6d64b19e2a3e9e3f234b9cbba4dbaed10c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:48.252700Z","signature_b64":"BNL2TzI7e2WUzIofKBo38Pub3dhOHd+s7vAew05PHNKJGfMtdWeeQ1e34ZMJcinvRS7W66ZflEFk2xU70UeZDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9902c7386fc1aa49878b90eaf76b732892bfbf1faaf009860a88d483e6fbeeba","last_reissued_at":"2026-05-18T00:36:48.252003Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:48.252003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graham's Tree Reconstruction Conjecture and a Waring-Type Problem on Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Anton Swifton, Bill Kay, Joshua Cooper","submitted_at":"2011-09-02T19:08:27Z","abstract_excerpt":"Suppose $G$ is a tree. 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