1. \begin{module}[id=bbt-size]
2. \importmodule[balanced-binary-trees]{balanced-binary-trees}
3. \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
4.  
5. \begin{frame}
6.  \frametitle{Size Lemma for Balanced Trees}
7.  \begin{itemize}
8.  \item
9.    \begin{assertion}[id=size-lemma,type=lemma]
10.    Let $G=\tup{V,E}$ be a \termref[cd=binary-trees]{balanced binary tree}
11.    of \termref[cd=graph-depth,name=vertex-depth]{depth}$n>i$, then the set
12.     $\defeq{\livar{V}i}{\setst{\inset{v}{V}}{\gdepth{v} = i}}$ of
13.    \termref[cd=graphs-intro,name=node]{nodes} at
14.    \termref[cd=graph-depth,name=vertex-depth]{depth} $i$ has
15.    \termref[cd=cardinality,name=cardinality]{cardinality} $\power2i$.
16.   \end{assertion}
17.  \item
18.    \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth $i$.}
19.      \begin{spfcases}{We have to consider two cases}
20.        \begin{spfcase}{$i=0$}
21.          \begin{spfstep}[display=flow]
22.            then $\livar{V}i=\set{\livar{v}r}$, where $\livar{v}r$ is the root, so
23.            $\eq{\card{\livar{V}0},\card{\set{\livar{v}r}},1,\power20}$.
24.          \end{spfstep}
25.        \end{spfcase}
26.        \begin{spfcase}{$i>0$}
27.          \begin{spfstep}[display=flow]
28.           then $\livar{V}{i-1}$ contains $\power2{i-1}$ vertexes
29.           \begin{justification}[method=byIH](IH)\end{justification}
30.          \end{spfstep}
31.          \begin{spfstep}
32.           By the \begin{justification}[method=byDef]definition of a binary
33.              tree\end{justification}, each $\inset{v}{\livar{V}{i-1}}$ is a leaf or has
34.            two children that are at depth $i$.
35.          \end{spfstep}
36.          \begin{spfstep}
37.           As $G$ is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and $\gdepth{G}=n>i$, $\livar{V}{i-1}$ cannot contain
38.            leaves.
39.          \end{spfstep}
40.          \begin{spfstep}[type=conclusion]
41.           Thus $\eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}}}},{\atimes[cdot]{2,\power2{i-1}}},\power2i}$.
42.          \end{spfstep}
43.        \end{spfcase}
44.      \end{spfcases}
45.    \end{sproof}
46.  \item
47.     \begin{assertion}[id=fbbt,type=corollary]  
48.      A fully balanced tree of depth $d$ has $\power2{d+1}-1$ nodes.
49.    \end{assertion}
50.  \item
51.      \begin{sproof}[for=fbbt,id=fbbt-pf]{}
52.        \begin{spfstep}
53.          Let $\defeq{G}{\tup{V,E}}$ be a fully balanced tree
54.        \end{spfstep}
55.        \begin{spfstep}
56.          Then $\card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1$.
57.        \end{spfstep}
58.      \end{sproof}
59.    \end{itemize}
60.  \end{frame}
61. \begin{note}
62.  \begin{omtext}[type=conclusion,for=binary-tree]
63.    This shows that balanced binary trees grow in breadth very quickly, a consequence of
64.    this is that they are very shallow (and this compute very fast), which is the essence of
65.    the next result.
66.  \end{omtext}
67. \end{note}
68. \end{module}
69.  
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