Skip to main content\(\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Pow}{\mathcal{P}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 5.6 Trees
Subsection 5.6.1 Terminology
Definition 5.6.1. Tree.
A graph is a
tree if and only if it is connected and contains no cycles.
Definition 5.6.2. Leaf.
A vertex in a tree is a
leaf if and only if it has degree one.
Definition 5.6.3. Spanning Tree.
A graph
\(G\) is a
spanning tree of a graph
H if and only if
\(G\) is a subgraph of
\(H\) that contains all the vertices of
\(H\) and is a tree.
Subsection 5.6.2 Practice
Checkpoint 5.6.4.
Checkpoint 5.6.5.
Draw all non-isomorphic trees on 3 vertices.
Checkpoint 5.6.6.
Prove that every tree with at least two vertices has at least one leaf.
Checkpoint 5.6.7.
Draw a tree with 7 vertices. Determine for what
\(n\) the tree is
\(n\)-connected.
Checkpoint 5.6.8.
Explain why for any tree removal of a leaf produces another tree.
Checkpoint 5.6.9.