~ According to (Fenton and Pfleeger, 1996), any tree impurity metric m should have the following properties:

a. m(G) = 0 if and only if G is a tree;

b. m(G1) > m(G2) if G1 differs from G2 only by the insertion of an extra arc;

c. For i = 1; 2 let Ai denote the number of arcs in Gi and Ni the number of nodes in Gi . Then if N1 > N2 and A1 N1 + 1 = A2 N2 + 1, then m(G1) < m(G2

d. For all graphs G, m(G)  m(KN ) = 1 where N = number of nodes of G and KN is the (undirected) complete graph of N nodes.
Give an intuitive rationale for these properties. Show that the tree impurity metric discussed in section 12.1.5 has these properties.