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Chapter 11: Q10E (page 755)

Draw the subtree of the tree in Exercise \(4\)that is rooted at

\(\begin{array}{l}{\bf{a}})a.\\b)c.\\c)e.\end{array}\)

Short Answer

Expert verified
  1. The root of the tree is the vertex at the top of the tree. Since, a is the root of the given tree, the subtree rooted at a is the entire tree itself.
  2. The subtree rooted at c contains the part of the given tree that contains the vertex c and all descendants of c. Note that the root of the subtree then becomes the vertex c and that c does not have any descendants.
  3. The subtree rooted at e contains the part of the given tree that contains the vertex e and all descendants of e. Note that the root of the subtree then becomes the vertex e.

Step by step solution

01

(a) The root of the tree

The root of the tree is the vertex at the top of the tree.

Since a is the root of the given tree, the subtree rooted at a is the entire tree itself.

02

(b) The subtree of c

The subtree rooted at c contains the part of the given tree that contains the vertex c and all descendants of c. Note that the root of the subtree then becomes the vertex c and that c does not have any descendants.

03

(c) The subtree of e

The subtree rooted at e contains the part of the given tree that contains the vertex e and all descendants of e. Note that the root of the subtree then becomes the vertex e.

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Most popular questions from this chapter

Sollin's algorithm produces a minimum spanning tree from a connected weighted simple graph \({\bf{G = (V,E)}}\) by successively adding groups of edges. Suppose that the vertices in \({\bf{V}}\) are ordered. This produces an ordering of the edges where \({\bf{\{ }}{{\bf{u}}_{\bf{0}}}{\bf{,}}{{\bf{v}}_{\bf{0}}}{\bf{\} }}\) precedes \({\bf{\{ }}{{\bf{u}}_{\bf{1}}}{\bf{,}}{{\bf{v}}_{\bf{1}}}{\bf{\} }}\) if \({{\bf{u}}_{\bf{0}}}\) precedes \({{\bf{u}}_{\bf{1}}}\) or if \({{\bf{u}}_{\bf{0}}}{\bf{ = }}{{\bf{u}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{0}}}\) precedes \({{\bf{v}}_{\bf{1}}}\). The algorithm begins by simultaneously choosing the edge of least weight incident to each vertex. The first edge in the ordering is taken in the case of ties. This produces a graph with no simple circuits, that is, a forest of trees (Exercise \({\bf{24}}\) asks for a proof of this fact). Next, simultaneously choose for each tree in the forest the shortest edge between a vertex in this tree and a vertex in a different tree. Again the first edge in the ordering is chosen in the case of ties. (This produces a graph with no simple circuits containing fewer trees than were present before this step; see Exercise \({\bf{24}}\).) Continue the process of simultaneously adding edges connecting trees until \({\bf{n - 1}}\) edges have been chosen. At this stage a minimum spanning tree has been constructed.

Show that the addition of edges at each stage of Sollin’s algorithm produces a forest.

Which connected simple graphs have exactly one spanning tree?

Show that every tree is a planar graph.

Which of these are well-formed formulae over the symbols \(\left\{ {{\bf{x,y,z}}} \right\}\) and the set of binary operators \(\left\{ {{\bf{ \ast , + ,}} \circ } \right\}\)?

  1. \({\bf{ \ast + + xyx}}\)
  2. \( \circ {\bf{xy \ast xz}}\)
  3. \({\bf{ \ast }} \circ {\bf{xz \ast \ast xy}}\)
  4. \({\bf{ \ast + }} \circ {\bf{xx}} \circ {\bf{xxx}}\)

Show that any well-formed formula in prefix notation over a set of symbols and a set of binary operators contains exactly one more symbol than the number of operators.
See all solutions

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