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Show full Prüfer code example

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Antti H S Laaksonen 2017-04-20 23:26:44 +03:00
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chapter22.tex
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@ -842,7 +842,8 @@ that removes $n-2$ leaves from the tree.
At each step, the leaf with the smallest label is removed, At each step, the leaf with the smallest label is removed,
and the label of its only neighbor is added to the code. and the label of its only neighbor is added to the code.
For example, the Prüfer code for For example, let us calculate the Prüfer code
of the following graph:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
\node[draw, circle] (1) at (2,3) {$1$}; \node[draw, circle] (1) at (2,3) {$1$};
@ -851,17 +852,62 @@ For example, the Prüfer code for
\node[draw, circle] (4) at (4,1) {$4$}; \node[draw, circle] (4) at (4,1) {$4$};
\node[draw, circle] (5) at (5.5,2) {$5$}; \node[draw, circle] (5) at (5.5,2) {$5$};
%\path[draw,thick,-] (1) -- (2);
%\path[draw,thick,-] (1) -- (3);
\path[draw,thick,-] (1) -- (4); \path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (3) -- (4); \path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4); \path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5); \path[draw,thick,-] (2) -- (5);
%\path[draw,thick,-] (4) -- (5);
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
is $[4,4,2]$, because we first remove
node 1, then node 3 and finally node 5. First we remove node 1 and add node 4 to the code:
\begin{center}
\begin{tikzpicture}[scale=0.9]
%\node[draw, circle] (1) at (2,3) {$1$};
\node[draw, circle] (2) at (4,3) {$2$};
\node[draw, circle] (3) at (2,1) {$3$};
\node[draw, circle] (4) at (4,1) {$4$};
\node[draw, circle] (5) at (5.5,2) {$5$};
%\path[draw,thick,-] (1) -- (4);
\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
\end{tikzpicture}
\end{center}
Then we remove node 3 and add node 4 to the code:
\begin{center}
\begin{tikzpicture}[scale=0.9]
%\node[draw, circle] (1) at (2,3) {$1$};
\node[draw, circle] (2) at (4,3) {$2$};
%\node[draw, circle] (3) at (2,1) {$3$};
\node[draw, circle] (4) at (4,1) {$4$};
\node[draw, circle] (5) at (5.5,2) {$5$};
%\path[draw,thick,-] (1) -- (4);
%\path[draw,thick,-] (3) -- (4);
\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
\end{tikzpicture}
\end{center}
Finally we remove node 4 and add node 2 to the code:
\begin{center}
\begin{tikzpicture}[scale=0.9]
%\node[draw, circle] (1) at (2,3) {$1$};
\node[draw, circle] (2) at (4,3) {$2$};
%\node[draw, circle] (3) at (2,1) {$3$};
%\node[draw, circle] (4) at (4,1) {$4$};
\node[draw, circle] (5) at (5.5,2) {$5$};
%\path[draw,thick,-] (1) -- (4);
%\path[draw,thick,-] (3) -- (4);
%\path[draw,thick,-] (2) -- (4);
\path[draw,thick,-] (2) -- (5);
\end{tikzpicture}
\end{center}
Thus, the Prüfer code of the graph is $[4,4,2]$.
We can construct a Prüfer code for any tree, We can construct a Prüfer code for any tree,
and more importantly, and more importantly,