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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="../../theme/default.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>