Question

Show that a center should be chosen as the root to producea rooted tree of minimal height from an unrooted tree.

Short answer

A tree is a connected graph having no simple circuits.

A rooted tree is a tree in which one vertex has been designated as the root and every edge is directed away from the root.

Step-by-step solution

Eccentricity of a vertex

A tree is a connected graph having no simple circuits.

A rooted tree is a tree in which one vertex has been designated as the root and every edge is directed away from the root.

The eccentricity of a vertex in an un rooted tree is the length of the longest simple path beginning at this vertex.

Vertex

A vertex is called a center if no vertex in the tree has smaller eccentricity than this vertex.

Because center is a vertex of a tree from which the length of the longest simple path is minimal, a center should be chosen as the root to produce rooted sub tree of minimal height from the given un rooted tree.