{"id":221491,"date":"2025-05-29T17:24:51","date_gmt":"2025-05-29T17:24:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221491"},"modified":"2025-05-29T17:24:53","modified_gmt":"2025-05-29T17:24:53","slug":"a-3-ary-tree-is-a-tree-where-every-internal-node-has-exactly-3-children","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/a-3-ary-tree-is-a-tree-where-every-internal-node-has-exactly-3-children\/","title":{"rendered":"A 3-ary tree is a tree where every internal node has exactly 3 children"},"content":{"rendered":"\n<p>A 3-ary tree is a tree where every internal node has exactly 3 children. Use induction to prove that the number of leaves in a 3-ary tree with n internal nodes is 2(n &#8211; 1) + 3.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To <strong>prove by induction<\/strong> that the number of leaves LL in a <strong>3-ary tree<\/strong> with nn internal nodes is: L=2(n\u22121)+3=2n+1L = 2(n &#8211; 1) + 3 = 2n + 1<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Base Case<\/strong>:<\/h3>\n\n\n\n<p>Let n=1n = 1 (a tree with one internal node).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since it&#8217;s a <strong>3-ary tree<\/strong>, the one internal node has exactly <strong>3 children<\/strong>.<\/li>\n\n\n\n<li>These 3 children must be <strong>leaves<\/strong>, because there are no other internal nodes.<\/li>\n<\/ul>\n\n\n\n<p>So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Number of leaves L=3L = 3<\/li>\n\n\n\n<li>Check the formula:<br>2n+1=2(1)+1=32n + 1 = 2(1) + 1 = 3 \u2714<\/li>\n<\/ul>\n\n\n\n<p>\u2705 Base case holds.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Inductive Hypothesis<\/strong>:<\/h3>\n\n\n\n<p>Assume the formula holds for some n=kn = k, i.e.,<br>A 3-ary tree with kk internal nodes has: L=2k+1&nbsp;leavesL = 2k + 1 \\text{ leaves}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Inductive Step<\/strong>:<\/h3>\n\n\n\n<p>We must prove that a tree with k+1k+1 internal nodes has: L=2(k+1)+1=2k+3&nbsp;leavesL = 2(k+1) + 1 = 2k + 3 \\text{ leaves}<\/p>\n\n\n\n<p>Start with the 3-ary tree of kk internal nodes and 2k+12k + 1 leaves.<br>Now, <strong>add 1 internal node<\/strong> by <strong>replacing one leaf<\/strong> with a new internal node having <strong>3 children<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>That removes <strong>1 leaf<\/strong> (the one we replaced)<\/li>\n\n\n\n<li>Adds <strong>3 new leaves<\/strong> (the children of the new internal node)<\/li>\n\n\n\n<li>So, net change in leaves: +2+2<br>(remove 1, add 3 \u2192 L\u2032=(2k+1\u22121+3)=2k+3L&#8217; = (2k + 1 &#8211; 1 + 3) = 2k + 3)<\/li>\n<\/ul>\n\n\n\n<p>Thus, a 3-ary tree with k+1k+1 internal nodes has 2k+3=2(k+1)+12k + 3 = 2(k+1) + 1 leaves.<\/p>\n\n\n\n<p>\u2705 Inductive step proven.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Conclusion<\/strong>:<\/h3>\n\n\n\n<p>By mathematical induction, in any <strong>3-ary tree<\/strong> with nn internal nodes, the number of <strong>leaves<\/strong> is: L=2n+1=2(n\u22121)+3L = 2n + 1 = 2(n &#8211; 1) + 3<\/p>\n\n\n\n<p><strong>Q.E.D.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-104.jpeg\" alt=\"\" class=\"wp-image-221492\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A 3-ary tree is a tree where every internal node has exactly 3 children. Use induction to prove that the number of leaves in a 3-ary tree with n internal nodes is 2(n &#8211; 1) + 3. The Correct Answer and Explanation is: To prove by induction that the number of leaves LL in a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-221491","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221491","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=221491"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221491\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}