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We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \\[2|E| \\prod_{v \\in V} (d(v) - t + 1)^{\\frac{(t-1)d(v)}{2|E|}}.\\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least\n  $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$.\n  This answers a question of Dellamonica et. al. (where the above statement was proved when $T$ is the path with three edges) while extending an old result of Erd\\H os and Simonovits."},"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":"1511.07274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-23T15:31:31Z","cross_cats_sorted":[],"title_canon_sha256":"ab9c466e52b481d42318e9a1f20de85b878b7c0afab209b0e3bf4d83c9843b80","abstract_canon_sha256":"46c4926ec5de13ccefcc267b582b826248fbdc1cfa73abcefb83df8daa7d5848"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:11.777176Z","signature_b64":"XWmUOfN9HrQKab3x2jZuPLwmcefHNPtkrJRZYzTCw1gw7ZEBD8BUaDWt1mv3xh95Ksuhv/hpmwJ+zh+DMj3HDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"136f22704342915ba1d4fe7be140a6bca22c1358c03ac4cbcebc759bfb4d117c","last_reissued_at":"2026-05-18T01:26:11.776614Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:11.776614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of trees in a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Jacques Verstraete","submitted_at":"2015-11-23T15:31:31Z","abstract_excerpt":"Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \\[2|E| \\prod_{v \\in V} (d(v) - t + 1)^{\\frac{(t-1)d(v)}{2|E|}}.\\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least\n  $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$.\n  This answers a question of Dellamonica et. al. 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