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     26 <h2>sTeX mode</h2>
     27 <form><textarea id="code" name="code">
     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 
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