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Given a family of real numbers parametrized by the $k$-subsets of $ \\{1,..., n\\}$, $\\{D_I\\}_{I \\in {\\{1,...,n\\} \\choose k}}$, we say that a weighted tree ${\\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\\cal T})=D_I$ for any $ I $. In [P-S] Pachter and Speyer proved that, if $3 \\leq k \\leq (n+1)/2$ and $"},"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":"1512.08494","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-28T20:03:21Z","cross_cats_sorted":[],"title_canon_sha256":"8dbf8ebb24e12634ad7571cbd49be9f468ecfa2cd8ff69ff92f11a5f858efb55","abstract_canon_sha256":"eb30e489eb2e2ee36867ad175016ce4b3c53bcf6c1e493d6078b5b896ec2bde2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:40.539487Z","signature_b64":"VUYaTrgNXsSugKej99XgZ8hOwPDvmn/gbd0+b5mRsiFpfwLm91iMtyx6C27VIXRuqrypUXo89DGlaYrhb4bPAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1725c0464c7b8ae3b161894d423b9105f2de787d25e0a1f85fa574f640034c6","last_reissued_at":"2026-05-18T01:23:40.538911Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:40.538911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Families of multiweights and pseudostars","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Agnese Baldisserri, Elena Rubei","submitted_at":"2015-12-28T20:03:21Z","abstract_excerpt":"Let ${\\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\\{i_1,..., i_k \\} \\subset \\{1,...,n\\}$,let $D_I ({\\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\\cal T})$ are called $k$-weights of ${\\cal T}$. 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