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1<!doctype html>
2<html>
3  <head>
4    <title>CodeMirror 2: sTeX mode</title>
5    <link rel="stylesheet" href="../../lib/codemirror.css">
6    <script src="../../lib/codemirror.js"></script>
7    <script src="stex.js"></script>
8    <link rel="stylesheet" href="stex.css">
9    <style>.CodeMirror {background: #f8f8f8;}</style>
10    <link rel="stylesheet" href="../../css/docs.css">
11  </head>
12  <body>
13    <h1>CodeMirror 2: sTeX mode</h1>
14     <form><textarea id="code" name="code">
15\begin{module}[id=bbt-size]
16\importmodule[balanced-binary-trees]{balanced-binary-trees}
17\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
18
19\begin{frame}
20  \frametitle{Size Lemma for Balanced Trees}
21  \begin{itemize}
22  \item
23    \begin{assertion}[id=size-lemma,type=lemma]
24    Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
25    of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
26     $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
27    \termref[cd=graphs-intro,name=node]{nodes} at
28    \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
29    \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
30   \end{assertion}
31  \item
32    \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
33      \begin{spfcases}{We have to consider two cases}
34        \begin{spfcase}{$i=0$}
35          \begin{spfstep}[display=flow]
36            then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
37            $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
38          \end{spfstep}
39        \end{spfcase}
40        \begin{spfcase}{$i>0$}
41          \begin{spfstep}[display=flow]
42           then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
43           \begin{justification}[method=byIH](IH)\end{justification}
44          \end{spfstep}
45          \begin{spfstep}
46           By the \begin{justification}[method=byDef]definition of a binary
47              tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
48            two children that are at depth $i$.
49          \end{spfstep}
50          \begin{spfstep}
51           As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
52            leaves.
53          \end{spfstep}
54          \begin{spfstep}[type=conclusion]
55           Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
56          \end{spfstep}
57        \end{spfcase}
58      \end{spfcases}
59    \end{sproof}
60  \item
61    \begin{assertion}[id=fbbt,type=corollary]   
62      A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
63    \end{assertion}
64  \item
65      \begin{sproof}[for=fbbt,id=fbbt-pf]{}
66        \begin{spfstep}
67          Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
68        \end{spfstep}
69        \begin{spfstep}
70          Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
71        \end{spfstep}
72      \end{sproof}
73    \end{itemize}
74  \end{frame}
75\begin{note}
76  \begin{omtext}[type=conclusion,for=binary-tree]
77    This shows that balanced binary trees grow in breadth very quickly, a consequence of
78    this is that they are very shallow (and this compute very fast), which is the essence of
79    the next result.
80  \end{omtext}
81\end{note}
82\end{module}
83
84%%% Local Variables:
85%%% mode: LaTeX
86%%% TeX-master: "all"
87%%% End: \end{document}
88</textarea></form>
89    <script>
90      var editor = CodeMirror.fromTextArea(document.getElementById("code"), {});
91    </script>
92
93    <p><strong>MIME types defined:</strong> <code>text/stex</code>.</p>
94
95  </body>
96</html>
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