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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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