-sTeX mode - -\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} - -%%% Local Variables: -%%% mode: LaTeX -%%% TeX-master: "all" -%%% End: \end{document} - - - - MIME types defined: text/x-stex. - - Parsing/Highlighting Tests: normal, verbose. - -
-
-
-