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<title>CodeMirror: sTeX mode</title>

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<a href="http://codemirror.net"><h1>CodeMirror</h1><img id=logo src="../../doc/logo.png"></a>

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<ul>

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<li><a href="../../doc/manual.html">Manual</a>

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</ul>

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<ul>

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<li><a href="../index.html">Language modes</a>

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</ul>

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</div>

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\begin{module}[id=bbt-size]

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\importmodule[balanced-binary-trees]{balanced-binary-trees}

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\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}

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\begin{frame}

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\frametitle{Size Lemma for Balanced Trees}

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\begin{itemize}

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\item

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\begin{assertion}[id=size-lemma,type=lemma]

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Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}

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of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set

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$\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of

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\termref[cd=graphs-intro,name=node]{nodes} at

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\termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has

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\termref[cd=cardinality,name=cardinality]{cardinality} $\power2ijQuery.

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\end{assertion}

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\item

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\begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $ijQuery.}

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\begin{spfcases}{We have to consider two cases}

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\begin{spfcase}{$i=0$}

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\begin{spfstep}[display=flow]

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then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so

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$\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}jQuery.

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\end{spfstep}

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\end{spfcase}

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\begin{spfcase}{$i>0$}

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\begin{spfstep}[display=flow]

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then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes

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\begin{justification}[method=byIH](IH)\end{justification}

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\end{spfstep}

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\begin{spfstep}

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By the \begin{justification}[method=byDef]definition of a binary

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tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has

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two children that are at depth $ijQuery.

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\end{spfstep}

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\begin{spfstep}

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As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain

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leaves.

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\end{spfstep}

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\begin{spfstep}[type=conclusion]

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Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}jQuery.

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\end{spfstep}

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\end{spfcase}

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\end{spfcases}

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\end{sproof}

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\item

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\begin{assertion}[id=fbbt,type=corollary]

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A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.

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\end{assertion}

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\item

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\begin{sproof}[for=fbbt,id=fbbt-pf]{}

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\begin{spfstep}

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Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree

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\end{spfstep}

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\begin{spfstep}

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Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1jQuery.

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\end{spfstep}

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\end{sproof}

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\end{itemize}

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\end{frame}

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\begin{note}

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\begin{omtext}[type=conclusion,for=binary-tree]

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This shows that balanced binary trees grow in breadth very quickly, a consequence of

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this is that they are very shallow (and this compute very fast), which is the essence of

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the next result.

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\end{omtext}

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\end{note}

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\end{module}

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%%% Local Variables:

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%%% mode: LaTeX

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%%% TeX-master: "all"

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%%% End: \end{document}

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</textarea></form>

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var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});

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</script>

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<p><strong>MIME types defined:</strong> <code>text/x-stex</code>.</p>

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<p><strong>Parsing/Highlighting Tests:</strong> <a href="../../test/index.html#stex_*">normal</a>, <a href="../../test/index.html#verbose,stex_*">verbose</a>.</p>

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</article>

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