From 01daa4d8d60e090b3e4cc826490e673cc94cfe70 Mon Sep 17 00:00:00 2001 From: Valentin Brandl Date: Mon, 12 Nov 2018 14:40:31 +0100 Subject: [PATCH] Add solution for dima u04 --- school/di-ma/uebung/04/04_1.tex | 70 ++++++++ school/di-ma/uebung/04/04_2.tex | 129 ++++++++++++++ school/di-ma/uebung/04/04_3.tex | 118 +++++++++++++ school/di-ma/uebung/04/04_4.tex | 77 +++++++++ school/di-ma/uebung/04/pascal_04.tex | 246 +++++++++++++++++++++++++++ 5 files changed, 640 insertions(+) create mode 100644 school/di-ma/uebung/04/04_1.tex create mode 100644 school/di-ma/uebung/04/04_2.tex create mode 100644 school/di-ma/uebung/04/04_3.tex create mode 100644 school/di-ma/uebung/04/04_4.tex create mode 100644 school/di-ma/uebung/04/pascal_04.tex diff --git a/school/di-ma/uebung/04/04_1.tex b/school/di-ma/uebung/04/04_1.tex new file mode 100644 index 0000000..6a1c396 --- /dev/null +++ b/school/di-ma/uebung/04/04_1.tex @@ -0,0 +1,70 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {4} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 4.1} +\begin{enumerate}[i)] + + \item Je zwei Koten in $G$ sind durch genau einen Pfad miteinander verbunden. + + $\Rightarrow G$ ist zusammenhängend $\Rightarrow G$ ist ein Baum (Lemma b) aus der Vorlesung) + + $\Rightarrow$ i) $\equiv$ iii) + + \item $G$ ist zusammenhängend und es gilt $|V| = |E| + 1$ + + + \item $G$ besitzt keinen einfachen Kreis und es gilt $|V| = |E| + 1$ + + $|V| = |E| + 1 \Rightarrow |V| - |E| = 1 \Rightarrow G$ hat eine Zusammenhangskomponente $\Rightarrow G$ ist + zusammenhängend. + + Kreisfrei und zusammenhängend $\Rightarrow G$ ist ein Baum + + $\Rightarrow$ ii) $\equiv$ iii) + +\end{enumerate} + +Da i) $\equiv$ iii) und ii) $\equiv$ iii) $\Rightarrow$ i) $\equiv$ ii) $\equiv$ iii) + +\end{document} + diff --git a/school/di-ma/uebung/04/04_2.tex b/school/di-ma/uebung/04/04_2.tex new file mode 100644 index 0000000..de2f991 --- /dev/null +++ b/school/di-ma/uebung/04/04_2.tex @@ -0,0 +1,129 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {4} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 4.2} +\begin{enumerate}[1.] + + \item Graph $G=(V,E)$ mit\\ + Knotenmenge: $V=\{a,b,c,d,e,f\}$\\ + Kantenmenge: $E= + \{ + \{a,b\}, + \{a,c\}, + \{a,d\}, + \{a,f\}, + \{b,f\}, + \{b,e\}, + \{c,d\}, + \{d,e\} + \} + $ + \\ + Adjazenz-Matrix:\\ + \begin{tabular}{c|ccccccc} + & a & b & c & d & e & f \\ + \hline + a & 0 & 1 & 1 & 1 & 0 & 1 \\ + b & 1 & 0 & 0 & 0 & 1 & 1 \\ + c & 1 & 0 & 0 & 1 & 0 & 0 \\ + d & 1 & 0 & 1 & 0 & 1 & 0 \\ + e & 0 & 1 & 0 & 1 & 0 & 0 \\ + f & 1 & 1 & 0 & 0 & 0 & 0 \\ + \end{tabular} + \\ + \\ + Inzidenz-Matrix:\\ + \begin{tabular}{c|ccccccccc} + a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ + b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ + c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ + d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ + e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ + f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ + \end{tabular} + + \item \begin{tikzpicture} + \def \n {5} + \def \radius {3cm} + \def \margin {8} % margin in angles, depends on the radius + \node[draw, circle] (a) at (3,2) {a}; + \node[draw, circle] (b) at (1,1) {b}; + \node[draw, circle] (c) at (2,1) {c}; + \node[draw, circle] (d) at (3,1) {d}; + \node[draw, circle] (f) at (4,1) {f}; + \node[draw, circle] (e) at (0,0) {e}; + + \draw (a)--(b); + \draw (a)--(c); + \draw (a)--(d); + \draw (a)--(f); + \draw (b)--(e); + \end{tikzpicture} + \\ + $G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$ + + \begin{tikzpicture} + \def \n {5} + \def \radius {3cm} + \def \margin {8} % margin in angles, depends on the radius + \node[draw, circle] (b) at (2,2) {b}; + \node[draw, circle] (a) at (1,1) {a}; + \node[draw, circle] (f) at (2,1) {f}; + \node[draw, circle] (e) at (3,1) {e}; + \node[draw, circle] (c) at (0,0) {c}; + \node[draw, circle] (d) at (2,0) {d}; + + \draw (b)--(a); + \draw (b)--(f); + \draw (b)--(e); + \draw (a)--(c); + \draw (a)--(d); + + \end{tikzpicture} + \\ + $G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$ + +\end{enumerate} + +\end{document} + diff --git a/school/di-ma/uebung/04/04_3.tex b/school/di-ma/uebung/04/04_3.tex new file mode 100644 index 0000000..2f7b40a --- /dev/null +++ b/school/di-ma/uebung/04/04_3.tex @@ -0,0 +1,118 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {4} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 4.3} +\begin{enumerate}[1.] + + \item Graph $G=(V,E)$ mit\\ + Knotenmenge: $V=\{1,2,3,4,5,6\}$\\ + Kantenmenge: $E= + \{ + (1,2), + (1,3), + (2,3), + (3,4), + (4,5), + (5,6), + (6,4) + \}$ + \\ + + Adjazenz-Matrix:\\ + \begin{tabular}{c|cccccc} + & 1 & 2 & 3 & 4 & 5 & 6 \\ + \hline + 1 & 0 & 1 & 1 & 0 & 0& 0 \\ + 2 & 0 & 0 & 1 & 0 & 0& 0 \\ + 3 & 0 & 0 & 0 & 1 & 0& 0 \\ + 4 & 0 & 0 & 0 & 0 & 1& 0 \\ + 5 & 0 & 0 & 0 & 0 & 0& 1 \\ + 6 & 0 & 0 & 0 & 1 & 0& 0 \\ + \end{tabular} + \\ + Inzidenz-Matrix:\\ + \begin{tabular}{c|ccccccc} + 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ + 2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ + 3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ + 4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ + 5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ + 6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ + \end{tabular} + + \item \begin{tikzpicture} + \def \n {5} + \def \radius {3cm} + \def \margin {8} % margin in angles, depends on the radius + \node[draw, circle] (1) at (0,1) {1}; + \node[draw, circle] (4) at (2,2) {4}; + \node[draw, circle] (6) at (2,0) {6}; + \node[draw, circle] (3) at (1,0) {3}; + \draw [->] + (1) edge (3) + (3) edge (4) + (6) edge (4); + \end{tikzpicture} + \\ + $V'=\{1,3,4,6\}$ + + \item \begin{tikzpicture} + \def \n {5} + \def \radius {3cm} + \def \margin {8} % margin in angles, depends on the radius + \node[draw, circle] (1) at (0,0) {1}; + \node[draw, circle] (3) at (1,0) {3}; + \node[draw, circle] (4) at (2,0) {4}; + \node[draw, circle] (5) at (3,0) {5}; + \draw [->] + (1) edge (3) + (3) edge (4) + (4) edge (5); + \end{tikzpicture} + \\ + $V'=\{1,3,4,5\}$ + +\end{enumerate} + +\end{document} + diff --git a/school/di-ma/uebung/04/04_4.tex b/school/di-ma/uebung/04/04_4.tex new file mode 100644 index 0000000..83d4b07 --- /dev/null +++ b/school/di-ma/uebung/04/04_4.tex @@ -0,0 +1,77 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {4} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 4.4} + +Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestens +$2 \cdot k + 2$ Blätter.\\ +\\ +IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben + +\begin{tikzpicture} + \node[draw,circle] (1) at (2,1) {1}; + \node[draw,circle] (2) at (0,0) {2}; + \node[draw,circle] (3) at (1,0) {3}; + \node[draw,circle] (4) at (3,0) {4}; + \node[draw,circle] (5) at (4,0) {5}; + + \draw (1)--(2); + \draw (1)--(3); + \draw (1)--(4); + \draw (1)--(5); +\end{tikzpicture} +\\ +$A(1) = 2 \cdot 1 + 2 = 4$\\ +\\ +\textbf{IV: $A(k) = 2 \cdot k +2$}\\ +\\ +IS: $k \rightarrow k+1$\\ +Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt +also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\ +\\ +$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$ +$= 2 \cdot (k+1)+2$\\ +q.e.d + +\end{document} + diff --git a/school/di-ma/uebung/04/pascal_04.tex b/school/di-ma/uebung/04/pascal_04.tex new file mode 100644 index 0000000..1c4aa9a --- /dev/null +++ b/school/di-ma/uebung/04/pascal_04.tex @@ -0,0 +1,246 @@ +\documentclass[10pt,a4paper]{article} +\usepackage[utf8x]{inputenc} +\usepackage{ucs} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{tikz} +\title{Blatt 04} +\begin{document} + +\maketitle +\newpage +\section{Aufgabe 1} +TODO +\section{Aufgabe 2} + +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (a) at (0,1) {a}; +\node[draw, circle] (b) at (1,2) {b}; +\node[draw, circle] (c) at (2,2) {c}; +\node[draw, circle] (d) at (3,1) {d}; +\node[draw, circle] (e) at (2,0) {e}; +\node[draw, circle] (f) at (1,0) {f}; +\draw (a)--(b); +\draw (a)--(c); +\draw (a)--(d); +\draw (a)--(f); +\draw (b)--(f); +\draw (b)--(e); +\draw (c)--(d); +\draw (d)--(e); +\end{tikzpicture} +\\ +\subsection{1.)} +Graph $G=(V,E)$ mit\\ +Knotenmenge: $V=\{a,b,c,d,e,f\}$\\ +Kantenmenge: $E= +\{ +\{a,b\}, +\{a,c\}, +\{a,d\}, +\{a,f\}, +\{b,f\}, +\{b,e\}, +\{c,d\}, +\{d,e\} +\} +$ +\\ +Adjazens-Matrix:\\ +\begin{tabular}{c|ccccccc} +& a & b & c & d & e & f \\ +\hline +a & 0 & 1 & 1 & 1 & 0 & 1 \\ +b & 1 & 0 & 0 & 0 & 1 & 1 \\ +c & 1 & 0 & 0 & 1 & 0 & 0 \\ +d & 1 & 0 & 1 & 0 & 1 & 0 \\ +e & 0 & 1 & 0 & 1 & 0 & 0 \\ +f & 1 & 1 & 0 & 0 & 0 & 0 \\ +\end{tabular} +\\ +\\ +Inz.-Matrix:\\ +\begin{tabular}{c|ccccccccc} +a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ +b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ +c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ +d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ +e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ +f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ +\end{tabular} + +\subsection{2.)} +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (a) at (3,2) {a}; +\node[draw, circle] (b) at (1,1) {b}; +\node[draw, circle] (c) at (2,1) {c}; +\node[draw, circle] (d) at (3,1) {d}; +\node[draw, circle] (f) at (4,1) {f}; +\node[draw, circle] (e) at (0,0) {e}; + +\draw (a)--(b); +\draw (a)--(c); +\draw (a)--(d); +\draw (a)--(f); +\draw (b)--(e); +\end{tikzpicture} +\\ +$G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$ + +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (b) at (2,2) {b}; +\node[draw, circle] (a) at (1,1) {a}; +\node[draw, circle] (f) at (2,1) {f}; +\node[draw, circle] (e) at (3,1) {e}; +\node[draw, circle] (c) at (0,0) {c}; +\node[draw, circle] (d) at (2,0) {d}; + +\draw (b)--(a); +\draw (b)--(f); +\draw (b)--(e); +\draw (a)--(c); +\draw (a)--(d); + +\end{tikzpicture} +\\ +$G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$ + + + + + + +\section{Aufgabe 3} + + +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (1) at (0,1) {1}; +\node[draw, circle] (2) at (1,2) {2}; +\node[draw, circle] (4) at (2,2) {4}; +\node[draw, circle] (5) at (3,1) {5}; +\node[draw, circle] (6) at (2,0) {6}; +\node[draw, circle] (3) at (1,0) {3}; + +\draw [->] +(1) edge (2) +(1) edge (3) +(2) edge (3) +(3) edge (4) +(4) edge (5) +(5) edge (6) +(6) edge (4); +\end{tikzpicture} + +\subsection{1.)} +Graph $G=(V,E)$ mit\\ +Knotenmenge: $V=\{1,2,3,4,5,6\}$\\ +Kantenmenge: $E= +\{ +(1,2), +(1,3), +(2,3), +(3,4), +(4,5), +(5,6), +(6,4) +\}$ +\\ + +Adjazens-Matrix:\\ +\begin{tabular}{c|cccccc} +& 1 & 2 & 3 & 4 & 5 & 6 \\ +\hline +1 & 0 & 1 & 1 & 0 & 0& 0 \\ +2 & 0 & 0 & 1 & 0 & 0& 0 \\ +3 & 0 & 0 & 0 & 1 & 0& 0 \\ +4 & 0 & 0 & 0 & 0 & 1& 0 \\ +5 & 0 & 0 & 0 & 0 & 0& 1 \\ +6 & 0 & 0 & 0 & 1 & 0& 0 \\ +\end{tabular} +\\ +Inzid.-Matrix:\\ +\begin{tabular}{c|ccccccc} +1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ +2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ +3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ +4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ +5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ +6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ +\end{tabular} +\subsection{2.)} +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (1) at (0,1) {1}; +\node[draw, circle] (4) at (2,2) {4}; +\node[draw, circle] (6) at (2,0) {6}; +\node[draw, circle] (3) at (1,0) {3}; +\draw [->] +(1) edge (3) +(3) edge (4) +(6) edge (4); +\end{tikzpicture} +\\ +$V'=\{1,3,4,6\}$\\ +\\ +\subsection{3.)} +\begin{tikzpicture} +\def \n {5} +\def \radius {3cm} +\def \margin {8} % margin in angles, depends on the radius +\node[draw, circle] (1) at (0,0) {1}; +\node[draw, circle] (3) at (1,0) {3}; +\node[draw, circle] (4) at (2,0) {4}; +\node[draw, circle] (5) at (3,0) {5}; +\draw [->] +(1) edge (3) +(3) edge (4) +(4) edge (5); +\end{tikzpicture} +\\ +$V'=\{1,3,4,5\}$\\ +\section{Aufgabe 4} + +Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestenz $2 \cdot k + 2$ Blätter.\\ +\\ +IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben + +\begin{tikzpicture} +\node[draw,circle] (1) at (2,1) {1}; +\node[draw,circle] (2) at (0,0) {2}; +\node[draw,circle] (3) at (1,0) {3}; +\node[draw,circle] (4) at (3,0) {4}; +\node[draw,circle] (5) at (4,0) {5}; + +\draw (1)--(2); +\draw (1)--(3); +\draw (1)--(4); +\draw (1)--(5); +\end{tikzpicture} +\\ +$A(1) = 2 \cdot 1 + 2 = 4$\\ +\\ +\textbf{IV: $A(k) = 2 \cdot k +2$}\\ +\\ +IS: $k \rightarrow k+1$\\ +Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\ +\\ +$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$ +$= 2 \cdot (k+1)+2$\\ +q.e.d +\end{document} \ No newline at end of file