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