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