 Rooted tree and binary tree  Theorem 5.19: A full binary tree with t leaves contains i=t-1 internal vertices.

 Theorem 5.20: Let T be a full binary tree. We denote the sum of length of simple paths from root to all internal vertices by I, and the sum of length of simple paths from root to all leaves by E. Then E=I+2i, where i is the number of internal vertices.  Proof: Let us apply induction on the number i of internal vertices.  E=2 and I=0 when i=1

 Suppose that result holds for i=k-1  For i=k, we chose internal vertex v so that its children are leaves. We have a new tree which is obtained by omitting edges of incident v and its children.  By the inductive hypothesis, E'=I'+2(k-1)  We denote the length of path from root to v by l.  E'=E- l-2, I'=I-l.  E= E'+l+2=I'+2(k-1)+l+2=I-l+2(k-1)+l+2=I+2k 。  Let T be a full m-ary tree. Then E=(m-1)I+mi, where i is the number of internal vertices.

5.6 Prefix codes and optimal tree  a b c d e   The set {00,110,010,10,01} is called code   ead or cc?  The string of e is prefix of string of c  c: 111  The set {00,110,111,10,01} is called prefix code

 Definition 31: Codes with this property which the bit string for a letter never occurs as the first part of the bit string for another letter are called prefix codes.  Theorem 5.21: We can construct a prefix code from any binary tree, and we can construct a binary tree from the prefix codes.  Proof: (1) We can construct a prefix code from any binary tree where the left edge at each internal vertex is labeled by 0 and the right edge by a 1 and where the leaves are labeled by characters  (2)We can construct a binary tree from the prefix codes

 Huffman algorithm:  Let a 1,a 2, ,a n be n vertex with weight w 1,w 2, ,w n and w 1  w 2  w n 。  F:=forest of n rooted tree each consisting of the single vertex a i with weight w i for i=1,2,...n.  While F is not a tree  Begin  Replace the rooted trees T and T’ of least weights from F with w(T)≥ w(T’) with a tree having a new root that has T as its left subtree and T’ as its right subtree. Assign w(T)+w(T’) as the weight of the new tree.  end

 Example: Find a optimal tree with weight 2,4,7,8,10,12

w(T)=2*4+4*4+7*3+ 12*2+8*2+10*2=105

 Theorem 5.22: Let T be built according to Huffman algorithm and leaves of T with weight w 1,w 2, ,w n. Then T is an optimal tree.  Proof: Let us apply induction on the number n of vertices.  n=2, The results holds Suppose that result holds for n=k-1 For n=k ， By the inductive hypothesis, Suppose that nodes in an optimal tree T have weights w 1 +w 2,w 3, ,w k. Then This is an optimal tree with weight w 1,w 2,w 3, ,w k if T’s leaf with weight w 1 +w 2 is replaced by subtree

 Lemma 5.1 Let T 1 be an optimal tree with weights w 1  w 2  w 3  w k. Then there is an optimal tree T 2 so that T 2 ’s two vertices with weights w 1 and w 2 are brother nodes.  Proof: Let T 1 be an optimal tree with weights w 1  w 2  w 3  w k  The two weights w a,w b are on lowest level and they are brothers. We denoted by v a and v b. Let v 0 be the father of v a, v b.  w a  w 1, w b  w 2.  Let l a be the length of the path from root to v a, and l b be the length of the path from root to v b.  l a =l b, l a  l 1, l a  l 2,  We obtain a new tree T 2 by exchanging from leaf v 1 to v a and from leaf v 2 to v b.  w(T 1 )-w(T 2 )=(w a -w 1 )(l a -l 1 )+ (w b -w 2 )(l a -l 2 )  0

 Lemma 5.2: Suppose that nodes of an optimal tree T have weights w 1 +w 2,w 3, ,w k. Then this is an optimal tree with weight w 1,w 2,w 3, ,w k if T*’s leaf with weight w 1 +w 2 is replaced by subtree Proof: By the Lemma 5.1,there is an optimal tree T 2 with weight w 1,w 2,w 3, ,w k so that T 2 ’s two vertices with weights w 1 and w 2 are brother nodes. Let l 1 be the length of the path from root to v 1 with weight w 1. Let T 2 * be a same tree as T 2 without leaf with weight w 1 and leaf with weight w 2, but with a having leaf of weight w 1 +w 2. w(T 2 *)=w(T 2 )-w 1 l 1 - w 2 l 1 +(w 1 +w 2 )(l 1 -1) Thus w(T 2 )=w(T 2 *)+w 1 +w 2 Let T* be a same tree as T without leaf with w 1 +w 2 but with subtree Suppose that T* is not an optimal tree.

 Theorem 5.22:Proof: Let us apply induction on the number n of vertices.  n=2, The results holds Suppose that result holds for n=k-1 For n=k ， By the inductive hypothesis, the tree with weight w 1 +w 2,w 3, ,w k by according to Huffman algorithm is an optimal tree. By lemma 5.2, this is an optimal tree with weight w 1,w 2,w 3, ,w k if T’s leaf with weight w 1 +w 2 is replaced by subtree

5.7 Transport Networks  Transport Networks  Definition 33: A transport network or a network, is a connected digraph N(V,E,C) with the following properties:  (1)N has no loop.  (2)There is a unique node s, the source, that has in- degree 0. And there is a unique node t, the sink, that has out-degree 0.  (3)The graph is labeled. The label, c ij on edge (i,j) is a nonnegative number called the capacity of the edge. Let C={c ij |(i,j)  E}.

 Definition 34: A flow in a network N(V,E,C) is a function that assigns to each edge (i,j) of N a nonnegative number f ij that does not exceed c ij. A conservation flow is a flow with the properties:  for each vertex other than the source and sink. The sum  is called the value of the conservation flow. A conservation flow f is called maximum flow, if v f ≥v f’ for any conservation flow f ’ of N.

 Definition 35: Let N be a network with the source node and the sink node. If P is a subset of V containing source s but not sink t then E(P,V-P)is called a cut separating s from t.  In effect, a cut does “cut” a digraph into two pieces, one containing the source and one containing the sink. If the edges of a cut were removed, it has not any paths from source to sink.

The capacity of a cut E(P,V-P)is, we denote by C(P,V-P). i.e.

 A cut E(P,V-P) of N is called minimum cut, if C(P,V-P)≤C(P',V-P') for any cut E(P',V-P') of N.  Theorem 5.23: For every conservation flow f and any cut E(P,V-P), the result holds: V f  C(P,V-P).  V f  C(P,V-P) ，  V f max  C min (P,V-P)

 Lemma 5.3: Let f be a conservation flow, E(P,V-P) be a cut. If V f =C(P,V-P) ， then V f max =V f ， C min (P,V-P)=C(P,V-P).  Proof: By the theorem 5.23,  Frod,Falkerson  1956

 Next: Transport Networks, 8.4 P307  Graph Matching 8.5 P315  Exercise: P  1. Let T be a full m-ary tree. We denote the sum of length of simple paths from root to all internal vertices by I, and the sum of length of simple paths from root to all leaves by E. Then E=(m-1)I+2i, (m-1)i=t-1, where i is the number of internal vertices and t is the number of leaves  2.(1)Find a optimal binary tree with weight 1,3,8,9,12, 15,16  (2)Construct a algorithms for optimal 3-ary tree  (3)Find a optimal 3-ary tree with weight 1,2,3,4,5,6,7,8,9  (4)Find a optimal 3-ary tree with weight 1,2,3,4,5,6,7,8,