&o We have a spanning tree s' which is obtained from SUek by omitting the edge e 令 Because w(e)≤w(e"),w(S")≤w(S), and the number of common edges of s' and T are added 令W(T≤w(S), 令 Suppose:w(S)<w(T) ☆ contradiction %o Rooted tree and binary tree .o Prefix codes and optimal tree❖ We have a spanning tree S’ which is obtained from S∪ek by omitting the edge e'. ❖ Because w(ek )≤w(e'), w(S')≤w(S), and the number of common edges of S’ and T are added 1. ❖ W(T)≤W(S), ❖ Suppose: W(S)<W(T) ❖ contradiction ❖ Rooted tree and binary tree ❖ Prefix codes and optimal tree