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  \begin{module}[id=bbt-size]
  \importmodule[balanced-binary-trees]{balanced-binary-trees}
  \importmodule[\KWARCslides{dmath/en/cardinality}]{cardinality}
  
  \begin{frame}
    \frametitle{Size Lemma for Balanced Trees}
    \begin{itemize}
    \item
      \begin{assertion}[id=size-lemma,type=lemma] 
      Let G=\tup{V,E} be a \termref[cd=binary-trees]{balanced binary tree} 
      of \termref[cd=graph-depth,name=vertex-depth]{depth}n>i, then the set
       \defeq{\livar{V}i}{\setst{\inset{v}{V}{\gdepth{v} = i} of
      \termref[cd=graphs-intro,name=node]{nodes} at 
      \termref[cd=graph-depth,name=vertex-depth]{depth} i has
      \termref[cd=cardinality,name=cardinality]{cardinality} \power2i.
     \end{assertion}
    \item
      \begin{sproof}[id=size-lemma-pf,proofend=,for=size-lemma]{via induction over the depth i.}
        \begin{spfcases}{We have to consider two cases}
          \begin{spfcase}{i=0}
            \begin{spfstep}[display=flow]
              then \livar{V}i=\set{\livar{v}r}, where \livar{v}r is the root, so
              \eq{\card{\livar{V}0},\card{\set{\livar{v}r},1,\power20}.
            \end{spfstep}
          \end{spfcase}
          \begin{spfcase}{i>0}
            \begin{spfstep}[display=flow]
             then \livar{V}{i-1} contains \power2{i-1} vertexes 
             \begin{justification}[method=byIH](IH)\end{justification}
            \end{spfstep}
            \begin{spfstep}
             By the \begin{justification}[method=byDef]definition of a binary
                tree\end{justification}, each \inset{v}{\livar{V}{i-1} is a leaf or has
              two children that are at depth i.
            \end{spfstep}
            \begin{spfstep}
             As G is \termref[cd=balanced-binary-trees,name=balanced-binary-tree]{balanced} and \gdepth{G}=n>i, \livar{V}{i-1} cannot contain
              leaves.
            \end{spfstep}
            \begin{spfstep}[type=conclusion]
             Thus \eq{\card{\livar{V}i},{\atimes[cdot]{2,\card{\livar{V}{i-1}},{\atimes[cdot]{2,\power2{i-1}},\power2i}.
            \end{spfstep}
          \end{spfcase}
        \end{spfcases}
      \end{sproof}
    \item 
      \begin{assertion}[id=fbbt,type=corollary]        
        A fully balanced tree of depth d has \power2{d+1}-1 nodes.
      \end{assertion}
    \item
        \begin{sproof}[for=fbbt,id=fbbt-pf]{}
          \begin{spfstep}
            Let \defeq{G}{\tup{V,E} be a fully balanced tree
          \end{spfstep}
          \begin{spfstep}
            Then \card{V}=\Sumfromto{i}1d{\power2i}= \power2{d+1}-1.
          \end{spfstep}
        \end{sproof}
      \end{itemize}
    \end{frame}
  \begin{note}
    \begin{omtext}[type=conclusion,for=binary-tree]
      This shows that balanced binary trees grow in breadth very quickly, a consequence of
      this is that they are very shallow (and this compute very fast), which is the essence of
      the next result.
    \end{omtext}
  \end{note}
  \end{module}
  
  
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(C) Æliens 04/09/2009

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