Chapter 11: Q28SE (page 805)
Is a tree necessarily a cactus?
Therefore, the tree is a cactus.
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Suppose that G is a directed graph with no circuits. Describe how depth-first search can be used to carry out a topological sort of the vertices of G.
Draw the subtree of the tree in Exercise \(4\)that is rooted at
\(\begin{array}{l}{\bf{a}})a.\\b)c.\\c)e.\end{array}\)
a) Define pre-order, in-order, and post-order tree traversal.
b) Give an example of pre-order, post-order, and in-order traversal of a binary tree of your choice with at least \(12\) vertices.
Draw a game tree for him if the starting position consists of three piles with one, two, and three stones, respectively. When drawing the tree represent by the same vertex symmetric positions that result from the same move. Find the value of each vertex of the game tree. Who wins the game if both players follow an optimal strategy?
What is wrong with the following “proof” using mathematical induction of the statement that every tree with \({\bf{n}}\) vertices has a path of length \({\bf{n - 1}}\). Basis step: Every tree with one vertex clearly has a path of length \(0\). Inductive step: Assume that a tree with \({\bf{n}}\) vertices has a path of length \({\bf{n - 1}}\), which has \({\bf{u}}\) as its terminal vertex. Add a vertex \({\bf{v}}\) and the edge from \({\bf{u}}\)to \({\bf{v}}\). The resulting tree has \({\bf{n + 1}}\) vertices and has a path of length \({\bf{n}}\). This completes the inductive step.
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