{"id":2369,"date":"2026-01-03T03:56:22","date_gmt":"2026-01-03T03:56:22","guid":{"rendered":"https:\/\/www.ignoubcamca.com\/forum\/?p=2369"},"modified":"2026-01-03T04:11:28","modified_gmt":"2026-01-03T04:11:28","slug":"we-can-test-whether-a-node-m-is-a-proper-ancestor-of-a-node-n-by-testing-whether-m-precedes-n-in-x-order-but-follows","status":"publish","type":"post","link":"https:\/\/www.ignoubcamca.com\/forum\/we-can-test-whether-a-node-m-is-a-proper-ancestor-of-a-node-n-by-testing-whether-m-precedes-n-in-x-order-but-follows\/","title":{"rendered":"We can test whether a node \u2018 m\u2019 is a proper ancestor of a node \u2018 n\u2019 by testing whether \u2018 m\u2019 precedes \u2018 n\u2019 in X-order but follows \u2018 n\u2019 in Y-order , where X and Y are chosen from {pre, post, in}. Determine all those pairs X and Y for which this statement holds."},"content":{"rendered":"\n<p>[Answer of Question 2] This question is about <strong>tree traversals<\/strong> and how <strong>ancestor\u2013descendant relationships<\/strong> are reflected in traversal orders.<\/p>\n\n\n\n<p>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A node <strong>m<\/strong> is a <strong>proper ancestor<\/strong> of node <strong>n<\/strong><\/li>\n\n\n\n<li>Test condition:<br><strong>m precedes n in X-order<\/strong><br><strong>m follows n in Y-order<\/strong><\/li>\n\n\n\n<li>Where <strong>X, Y <\/strong><strong>\u2208 {preorder, postorder, inorder}<\/strong><\/li>\n<\/ul>\n\n\n\n<p>We must find <strong>all valid (X, Y) pairs<\/strong> for which this condition is <strong>always true<\/strong>.<\/p>\n\n\n\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\">Key traversal properties (very important)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. Preorder (Pre)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Visit <strong>root before children<\/strong><\/li>\n\n\n\n<li><strong>Ancestor always appears before descendant<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. Postorder (Post)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Visit <strong>root after children<\/strong><\/li>\n\n\n\n<li><strong>Ancestor always appears after descendant<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. Inorder (In)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Root appears <strong>between left and right subtrees<\/strong><\/li>\n\n\n\n<li><strong>No consistent ordering<\/strong> between ancestor and descendant in general<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Analyze all possible (X, Y) combinations<\/h2>\n\n\n\n<p>We want:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>m precedes n in X<\/strong><\/li>\n\n\n\n<li><strong>m follows n in Y<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Case 1: X = Preorder<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ancestor <strong>always precedes<\/strong> descendant \u2705<\/li>\n<\/ul>\n\n\n\n<p>Now Y must be a traversal where ancestor <strong>follows<\/strong> descendant.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Y = Postorder<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ancestor <strong>always follows<\/strong> descendant \u2705<br>\u2714 <strong>Valid pair: (Pre, Post)<\/strong><\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Y = Inorder<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inorder does <strong>not guarantee<\/strong> ancestor after descendant \u274c<br>\u2718 Not valid<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Case 2: X = Inorder<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inorder does <strong>not guarantee<\/strong> ancestor before descendant \u274c<br>\u2718 No pair starting with Inorder is valid<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Case 3: X = Postorder<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ancestor appears <strong>after<\/strong> descendant \u274c<br>\u2718 Cannot satisfy \u201cm precedes n\u201d<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Final Answer<\/h2>\n\n\n\n<p>The <strong>only pair (X, Y)<\/strong> for which the statement holds is:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>=<\/mo><mtext>Preorder<\/mtext><mo separator=\"true\">,<\/mo><mtext>\u2005\u200a<\/mtext><mi>Y<\/mi><mo>=<\/mo><mtext>Postorder<\/mtext><mo stretchy=\"false\">)<\/mo><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math><\/p>\n","protected":false},"excerpt":{"rendered":"<p>[Answer of Question 2] This question is about tree traversals and how ancestor\u2013descendant relationships are reflected in traversal orders. We are given: We must find all valid (X, Y) pairs for which this condition is always true.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29,30,35],"tags":[],"class_list":["post-2369","post","type-post","status-publish","format-standard","hentry","category-bca-assignment","category-january-2026","category-mcs-208"],"_links":{"self":[{"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/posts\/2369","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/comments?post=2369"}],"version-history":[{"count":2,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/posts\/2369\/revisions"}],"predecessor-version":[{"id":2378,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/posts\/2369\/revisions\/2378"}],"wp:attachment":[{"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/media?parent=2369"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/categories?post=2369"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ignoubcamca.com\/forum\/wp-json\/wp\/v2\/tags?post=2369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}