Exercise 10.

Consider the tree drawn below.

A labeled tree with nine vertices labeled a through h. Vertices a, b, e, f, and i are on a single row, adjacent on a path in that order from left to right. Vertex b is adjacent to c and d above it. Vertex f is adjacent to g and h about it.

  1. Suppose we designate vertex \(e\) as the root. List the children, parents and siblings of each vertex. Does any vertex other than \(e\) have grandchildren?

  2. Suppose \(e\) is not chosen as the root. Does our choice of root vertex change the number of children \(e\) has? The number of grandchildren? How many are there of each?

  3. In fact, pick any vertex in the tree and suppose it is not the root. Explain why the number of children of that vertex does not depend on which other vertex is the root.

  4. Does the previous part work for other trees? Give an example of a different tree for which it holds. Then either prove that it always holds or give an example of a tree for which it doesn't.

Hint.
in-context