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We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $\\beta(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n = \\beta^{-1} n \\ln(n /\\ln 4) + (\\ln 2)/(2 \\beta)$. We prove that $2^{-n} Z_n(T_n + n \\tau)$,"},"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":"0807.1750","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-bio.CB","submitted_at":"2008-07-10T22:00:46Z","cross_cats_sorted":["math-ph","math.MP","q-bio.QM","stat.OT"],"title_canon_sha256":"ad32a7f62373bb3f16b1eee434e76853394835ea1aa2dbdf020632eab8726c61","abstract_canon_sha256":"a51e34712ce0511952c5632e475f27fc96b0a093408c43b875831eed36a75cad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:59.455420Z","signature_b64":"FD6HyopBlkLYK53imd32Zt1UFT7B8Y9Yac/AlwWTeS3YE99QeyZ/XCApNeriS+lu1unx7t6F0fq8mOooZE8nCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"744de8331066e35ba77452d7cde9aec9fd1739b00fb2acb8e097729d36def0c0","last_reissued_at":"2026-05-18T03:48:59.454731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:59.454731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","q-bio.QM","stat.OT"],"primary_cat":"q-bio.CB","authors_text":"B. F. Svaiter, C. Landim, R. D. Portugal","submitted_at":"2008-07-10T22:00:46Z","abstract_excerpt":"Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\\ge 1$ and $\\beta>0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $\\beta(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n = \\beta^{-1} n \\ln(n /\\ln 4) + (\\ln 2)/(2 \\beta)$. 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