How to prove that height of a rooted tree is an invariant under isomorphism?












0












$begingroup$


Intuitively, if we're restricted to mapping the root of $T_1$ to the root of $T_2$ that makes the structure of the two trees rigid. And since under isomorphism the number of vertices are equal, height must also remain the same. But how can this be made into a formal proof? Perhaps some sort of a contradiction?










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Intuitively, if we're restricted to mapping the root of $T_1$ to the root of $T_2$ that makes the structure of the two trees rigid. And since under isomorphism the number of vertices are equal, height must also remain the same. But how can this be made into a formal proof? Perhaps some sort of a contradiction?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Intuitively, if we're restricted to mapping the root of $T_1$ to the root of $T_2$ that makes the structure of the two trees rigid. And since under isomorphism the number of vertices are equal, height must also remain the same. But how can this be made into a formal proof? Perhaps some sort of a contradiction?










      share|cite|improve this question









      $endgroup$




      Intuitively, if we're restricted to mapping the root of $T_1$ to the root of $T_2$ that makes the structure of the two trees rigid. And since under isomorphism the number of vertices are equal, height must also remain the same. But how can this be made into a formal proof? Perhaps some sort of a contradiction?







      discrete-mathematics graph-theory computer-science trees






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 4 '18 at 22:55









      alwaysiamcaesaralwaysiamcaesar

      525




      525






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Like you note, the isomorphism must associate the two roots. Additionally, it sends leaves to leaves. Finally, it preserves paths; any path in $T_1$ is still a path in $T_2$, after applying the isomorphism (and vice versa). The height is just the length of the longest path from the root to a leaf; since all root-to-leaf paths in $T_1$ are isomorphic to such a path in $T_2$ of the same length, we have $h(T_1) leq h(T_2)$; the same argument works the other way as well.






          share|cite|improve this answer











          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026296%2fhow-to-prove-that-height-of-a-rooted-tree-is-an-invariant-under-isomorphism%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Like you note, the isomorphism must associate the two roots. Additionally, it sends leaves to leaves. Finally, it preserves paths; any path in $T_1$ is still a path in $T_2$, after applying the isomorphism (and vice versa). The height is just the length of the longest path from the root to a leaf; since all root-to-leaf paths in $T_1$ are isomorphic to such a path in $T_2$ of the same length, we have $h(T_1) leq h(T_2)$; the same argument works the other way as well.






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              Like you note, the isomorphism must associate the two roots. Additionally, it sends leaves to leaves. Finally, it preserves paths; any path in $T_1$ is still a path in $T_2$, after applying the isomorphism (and vice versa). The height is just the length of the longest path from the root to a leaf; since all root-to-leaf paths in $T_1$ are isomorphic to such a path in $T_2$ of the same length, we have $h(T_1) leq h(T_2)$; the same argument works the other way as well.






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                Like you note, the isomorphism must associate the two roots. Additionally, it sends leaves to leaves. Finally, it preserves paths; any path in $T_1$ is still a path in $T_2$, after applying the isomorphism (and vice versa). The height is just the length of the longest path from the root to a leaf; since all root-to-leaf paths in $T_1$ are isomorphic to such a path in $T_2$ of the same length, we have $h(T_1) leq h(T_2)$; the same argument works the other way as well.






                share|cite|improve this answer











                $endgroup$



                Like you note, the isomorphism must associate the two roots. Additionally, it sends leaves to leaves. Finally, it preserves paths; any path in $T_1$ is still a path in $T_2$, after applying the isomorphism (and vice versa). The height is just the length of the longest path from the root to a leaf; since all root-to-leaf paths in $T_1$ are isomorphic to such a path in $T_2$ of the same length, we have $h(T_1) leq h(T_2)$; the same argument works the other way as well.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 5 '18 at 7:36

























                answered Dec 5 '18 at 7:14









                plattyplatty

                3,370320




                3,370320






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026296%2fhow-to-prove-that-height-of-a-rooted-tree-is-an-invariant-under-isomorphism%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    How to change which sound is reproduced for terminal bell?

                    Can I use Tabulator js library in my java Spring + Thymeleaf project?

                    Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents