{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:akron1365772885"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:akron1365772885","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"Painted Trees and Pterahedra","abstract":"Associahedra can be realized by taking the convex hull of coordinates derived from binary trees. Similarly, permutahedra can be found using leveled trees. In this paper we will introduce a new type of painted tree, (T ◦ Y)<sub><i>n</i></sub> where <i>n</i> is the number of interior nodes. We create these painted trees by composing binary trees on leveled trees. We define a coordinate system on these trees and take the convex hull of these points. We explore the resulting polytope and prove, using a bijection to tubings, that for <i>n</i> ≤ 4 the poset of the painted face trees with <i>n</i>+1 leaves is isomorphic to the face poset of an <i>n</i>-dimensional polytope, specifically KF<sub>1,<i>n</i></sub>, the graph-associahedron for a fan graph, F<sub>1,<i>n</i></sub>.","abstract_html":"Associahedra can be realized by taking the convex hull of coordinates derived from binary trees. Similarly, permutahedra can be found using leveled trees. In this paper we will introduce a new type of painted tree, (T ◦ Y)&lt;sub&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sub&gt; where &lt;i&gt;n&lt;/i&gt; is the number of interior nodes. We create these painted trees by composing binary trees on leveled trees. We define a coordinate system on these trees and take the convex hull of these points. We explore the resulting polytope and prove, using a bijection to tubings, that for &lt;i&gt;n&lt;/i&gt; ≤ 4 the poset of the painted face trees with &lt;i&gt;n&lt;/i&gt;+1 leaves is isomorphic to the face poset of an &lt;i&gt;n&lt;/i&gt;-dimensional polytope, specifically KF&lt;sub&gt;1,&lt;i&gt;n&lt;/i&gt;&lt;/sub&gt;, the graph-associahedron for a fan graph, F&lt;sub&gt;1,&lt;i&gt;n&lt;/i&gt;&lt;/sub&gt;.","abstract_has_math":false,"creators":["Berry, Lisa Tredway"],"institution":"University of Akron","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Forcey, Stefan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-21","date_published":"2013-05-21","updated_at":"2026-07-24T03:37:31Z","subjects":["Mathematics","painted trees","polytope","pterahedra","graph-associahedra","tubings"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=akron1365772885","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Forcey, Stefan"]},{"key":"dc:creator","label":"Author","values":["Berry, Lisa Tredway"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-21"]},{"key":"dc:publisher","label":"Institution","values":["University of Akron / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Akron"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","painted trees","polytope","pterahedra","graph-associahedra","tubings"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://rave.ohiolink.edu/etdc/view?acc_num=akron1365772885"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Associahedra can be realized by taking the convex hull of coordinates derived from binary trees. Similarly, permutahedra can be found using leveled trees. In this paper we will introduce a new type of painted tree, (T ◦ Y)<sub><i>n</i></sub> where <i>n</i> is the number of interior nodes. We create these painted trees by composing binary trees on leveled trees. We define a coordinate system on these trees and take the convex hull of these points. We explore the resulting polytope and prove, using a bijection to tubings, that for <i>n</i> ≤ 4 the poset of the painted face trees with <i>n</i>+1 leaves is isomorphic to the face poset of an <i>n</i>-dimensional polytope, specifically KF<sub>1,<i>n</i></sub>, the graph-associahedron for a fan graph, F<sub>1,<i>n</i></sub>."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.111","9.69 MB"]},{"key":"dc:title","label":"Title","values":["Painted Trees and Pterahedra"]}]}],"canonical_facts":{"dc:contributor":["Forcey, Stefan"],"dc:creator":["Berry, Lisa Tredway"],"dc:date":["2013-05-21"],"dc:description":["Associahedra can be realized by taking the convex hull of coordinates derived from binary trees. Similarly, permutahedra can be found using leveled trees. In this paper we will introduce a new type of painted tree, (T ◦ Y)<sub><i>n</i></sub> where <i>n</i> is the number of interior nodes. We create these painted trees by composing binary trees on leveled trees. We define a coordinate system on these trees and take the convex hull of these points. We explore the resulting polytope and prove, using a bijection to tubings, that for <i>n</i> ≤ 4 the poset of the painted face trees with <i>n</i>+1 leaves is isomorphic to the face poset of an <i>n</i>-dimensional polytope, specifically KF<sub>1,<i>n</i></sub>, the graph-associahedron for a fan graph, F<sub>1,<i>n</i></sub>."],"dc:format":["application/pdf","p.111","9.69 MB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=akron1365772885"],"dc:language":["English"],"dc:publisher":["University of Akron / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics","painted trees","polytope","pterahedra","graph-associahedra","tubings"],"dc:title":["Painted Trees and Pterahedra"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["University of Akron"]},"updated_at":"2026-07-24T03:37:31Z"}