91Ó°ÊÓ

a) Draw the graphs of all nonisomorphic trees on six vertices. b) How many isomers does hexane \(\left(\mathrm{C}_{6} \mathrm{H}_{14}\right.\) ) have?

Construct an optimal prefix code for the symbols \(a, b, c, \ldots, i, j\) that occur (in a given sample) with respective frequencies \(78,16,30,35,125,31,20,50,80,3\).

How many leaves does a full binary tree have if its height is, (a) \(3 ?\) (b) \(7 ?\) (c) \(12 ?\) (d) \(h\) ?

Let \(T=(V, E)\) be a rooted tree ordered by a universal address system. (a) If vertex \(v\) in \(T\) has address 2.1.3.6, what is the smallest number of siblings that \(v\) must have? (b) For the vertex \(v\) in part (a), find the address of its parent. (c) How many ancestors does the vertex \(v\) in part (a) have? (d) With the presence of \(v\) in \(T\), what other addresses must there be in the system?

Let \(T=(V, E)\) be a complete \(m\)-ary tree of height \(h\). This tree is called a full \(m\)-ary tree if all of its leaves are at level \(h\). If \(T\) is a full \(m\)-ary tree with height 7 and 279,936 leaves, how many internal vertices are there in \(T\) ?

a) If a tree has four vertices of degree 2 , one vertex of degree 3, two of degree 4, and one of degree 5 , how many pendant vertices does it have? b) If a tree \(T=(V, E)\) has \(v_{2}\) vertices of degree \(2, v_{3}\) vertices of degree \(3, \ldots\), and \(v_{m}\) vertices of degree \(m\), what are \(|V|\) and \(|E| ?\)

a) A complete ternary (or 3-ary) tree \(T=(V, E)\) has 34 internal vertices. How many edges does \(T\) have? How many leaves? b) How many internal vertices does a complete 5 -ary tree with 817 leaves have?

What is the maximum number of internal vertices that a complete quaternary tree of height 8 can have? What is the number for a complete \(m\)-ary tree of height \(h\) ?