Section 1
Introduction
Which of the graphs are trees? Explain.
For which values of $n$ is the complete graph on $n$ vertices a tree?
For which values of $m$ and $n$ is the complete bipartite graph on $m$ and $n$ vertices a tree?
For which values of $n$ is the $n$ -cube a tree?
Find the level of each vertex in the tree shown.
Find the height of the tree of Exercise 8 .
Draw the tree $T$ of Figure 9.1 .5 as a rooted tree with $a$ as root. What is the height of the resulting tree?
Draw the tree $T$ of Figure 9.1 .5 as a rooted tree with $b$ as root. What is the height of the resulting tree?
Give an example similar to Example 9.1 .5 of a tree that is used to specify hierarchical relationships.
Give an example different from Example 9.1 .7 of a hierarchical definition tree.
Decode each bit string using the Huffiman code given. $$011000010$$
Decode each bit string using the Huffiman code given.$$01111001001110$$
Decode each bit string using the Huffiman code given.$$01110100110$$
Decode each bit string using the Huffiman code given.$$1110011101001111$$
Encode each word using the preceding Huffiman code.DEN
Encode each word using the preceding Huffiman code.LEADEN
Encode each word using the preceding Huffiman code.NEED
Encode each word using the preceding Huffiman code.PENNED
What factors in addition to the amount of memory used should be considered when choosing a code, such as ASCII or a Huffman code, to represent characters in a computer?
What techniques in addition to the use of Huffman codes might be used to save memory when storing text?
Construct an optimal Huffman code for the set of letters in the table.$$\begin{array}{cc|cc}\hline \text { Letter } & \text { Frequency } & \text { Letter } & \text { Frequency } \\\hline \alpha & 5 & \delta & 11 \\\beta & 6 & \varepsilon & 20 \\\gamma & 6 & & \\\hline\end{array}$$
Construct an optimal Huffman code for the set of letters in the table.$$\begin{array}{cr|cr}\hline \text { Letter } & \text { Frequency } & \text { Letter } & \text { Frequency } \\\hline \text { I } & 7.5 & \text { C } & 5.0 \\\text { U } & 20.0 & \text { H } & 10.0 \\\text { B } & 2.5 & \text { M } & 2.5 \\\text { S } & 27.5 & \text { P } & 25.0 \\\hline\end{array}$$
Use the code developed in Exercise 25 to encode the following words (which have frequencies consistent with the table of Exercise 25 ):BUS, CUPS, MUSH, PUSS, SIP, PUSH,CUSS, HIP, PUP, PUPS, HIPS.
Construct two optimal Huffman coding trees for the table of Exercise 24 of different heights.
Construct an optimal Huffman code for the set of letters in the table$$\begin{array}{cc|cc}\hline \text { Letter } & \text { Frequency } & \text { Letter } & \text { Frequency } \\\hline a & 2 & d & 8 \\b & 3 & e & 13 \\c & 5 & f & 21 \\\hline\end{array}$$
Professor Ter A. Byte needs to store text made up of the characters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E},$ which occur with the following frequencies:$$\begin{array}{cc|cc}\hline \text { Character } & \text { Frequency } & \text { Character } & \text { Frequency } \\\hline \text { A } & 6 & \text { D } & 2 \\\text { B } & 2 & \text { E } & 8 \\\text { C } & 3 & & \\\hline\end{array}$$Professor Byte suggests using the variable-length codes$$\begin{array}{cr}\hline \text { Character } & \text { Code } \\\hline \text { A } & 1 \\\text { B } & 00 \\\text { C } & 01 \\\text { D } & 10 \\\text { E } & 0\end{array}$$which, he argues, store the text in less space than that used by an optimal Huffman code. Is the professor correct? Explain.
Show that a tree is a planar graph.
Show that any tree with two or more vertices has a vertex of degree 1 .
Show that a tree is a bipartite graph.
Show that the vertices of a tree can be colored with two colors so that each edge is incident on vertices of different colors.
Find the eccentricity of each vertex in the tree of Figure $9.1 .5 .$
Find the center(s) of the tree of Figure $9.1 .5 .$
Show that a tree has either one or two centers.
Show that if a tree has two centers they are adjacent.
Define the radius $r$ of a tree using the concepts of eccentricity and center. The diameter $d$ of any graph was defined before Exercise $71,$ Section $8.2 .$ Is it always true, according to your definition of radius, that $2 r=d ?$ Explain.
Give an example of a tree $T$ that does not satisfy the following property: If $v$ and $w$ are vertices in $T,$ there is a unique path from $v$ to $w$