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  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}
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