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26<h2>sTeX mode</h2>
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28\begin{module}[id=bbt-size]
29\importmodule[balanced-binary-trees]{balanced-binary-trees}
30\importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
31
32\begin{frame}
33 \frametitle{Size Lemma for Balanced Trees}
34 \begin{itemize}
35 \item
36 \begin{assertion}[id=size-lemma,type=lemma]
37 Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
38 of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
39 $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
40 \termref[cd=graphs-intro,name=node]{nodes} at
41 \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
42 \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
43 \end{assertion}
44 \item
45 \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
46 \begin{spfcases}{We have to consider two cases}
47 \begin{spfcase}{$i=0$}
48 \begin{spfstep}[display=flow]
49 then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
50 $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
51 \end{spfstep}
52 \end{spfcase}
53 \begin{spfcase}{$i>0$}
54 \begin{spfstep}[display=flow]
55 then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
56 \begin{justification}[method=byIH](IH)\end{justification}
57 \end{spfstep}
58 \begin{spfstep}
59 By the \begin{justification}[method=byDef]definition of a binary
60 tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
61 two children that are at depth $i$.
62 \end{spfstep}
63 \begin{spfstep}
64 As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
65 leaves.
66 \end{spfstep}
67 \begin{spfstep}[type=conclusion]
68 Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
69 \end{spfstep}
70 \end{spfcase}
71 \end{spfcases}
72 \end{sproof}
73 \item
74 \begin{assertion}[id=fbbt,type=corollary]
75 A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
76 \end{assertion}
77 \item
78 \begin{sproof}[for=fbbt,id=fbbt-pf]{}
79 \begin{spfstep}
80 Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
81 \end{spfstep}
82 \begin{spfstep}
83 Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
84 \end{spfstep}
85 \end{sproof}
86 \end{itemize}
87 \end{frame}
88\begin{note}
89 \begin{omtext}[type=conclusion,for=binary-tree]
90 This shows that balanced binary trees grow in breadth very quickly, a consequence of
91 this is that they are very shallow (and this compute very fast), which is the essence of
92 the next result.
93 \end{omtext}
94\end{note}
95\end{module}
96
97%%% Local Variables:
98%%% mode: LaTeX
99%%% TeX-master: "all"
100%%% End: \end{document}
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