平衡树实现
•定义1（AVL树）：An empty binary tree is height balanced. If T is a nonempty binary tree with TL and TR as its left and right subtrees, then T is height balanced iff
• (1) TL and TR are height balanced, and
• (2) | hL - hR | £ 1 where hL and hR are the heights of TL and TR , respectively.
•定义2（平衡因子）：The balance factor BF( node ) = hL - hR . In an AVL tree, BF( node ) = -1, 0, or 1.
AVL树的最少节点数
•我们把必须重新平衡的节点叫做α，节点α的两棵子树的高度差为2。容易看出，这种不平衡可以出现在以下四种情况中：
•1、对α的左儿子的左子树进行一次插入。
•2、对α的左儿子的右子树进行一次插入。
•3、对α的右儿子的左子树进行一次插入。
•4、对α的右儿子的右子树进行一次插入。
•情况1和4是关于α对称的，情况2和3也是关于α对称的。
1 static Position 2 SingleRotateWithLeft( Position K2 ) 3 { 4 Position K1; 5 6 K1 = K2->Left; 7 K2->Left = K1->Right; 8 K1->Right = K2; 9 10 K2->Height = Max( Height( K2->Left ), Height( K2->Right ) ) + 1; 11 K1->Height = Max( Height( K1->Left ), K2->Height ) + 1; 12 13 return K1; /* New root */ 14 }
SingleRotateWithLeft单旋转例程
1 static Position 2 DoubleRotateWithLeft( Position K3 ) 3 { 4 /* Rotate between K1 and K2 */ 5 K3->Left = SingleRotateWithRight( K3->Left ); 6 7 /* Rotate between K3 and K2 */ 8 return SingleRotateWithLeft( K3 ); 9 }
DoubleRotateWithLeft双旋转例程
avltree.h
1 /*平衡树的声明*/ 2 typedef int ElementType; 3 4 /* START: fig4_35.txt */ 5 #ifndef _AvlTree_H 6 #define _AvlTree_H 7 8 struct AvlNode; 9 typedef struct AvlNode *Position; 10 typedef struct AvlNode *AvlTree; 11 12 AvlTree MakeEmpty( AvlTree T ); 13 Position Find( ElementType X, AvlTree T ); 14 Position FindMin( AvlTree T ); 15 Position FindMax( AvlTree T ); 16 AvlTree Insert( ElementType X, AvlTree T ); 17 AvlTree Delete( ElementType X, AvlTree T ); 18 ElementType Retrieve( Position P ); 19 20 //以下是补充 21 void PrintTree(AvlTree T); 22 AvlTree Remove(const ElementType X, AvlTree T); 23 AvlTree Remove( ElementType X, AvlTree T ); 24 25 #endif /* _AvlTree_H */ 26 /* END */
avltree.c
1 #include "avltree.h" 2 #include <stdlib.h> 3 #include "fatal.h" 4 5 struct AvlNode 6 { 7 ElementType Element; 8 AvlTree Left; 9 AvlTree Right; 10 int Height; 11 }; 12 13 /*置空平衡树*/ 14 AvlTree 15 MakeEmpty( AvlTree T ) 16 { 17 if( T != NULL ) 18 { 19 MakeEmpty( T->Left ); 20 MakeEmpty( T->Right ); 21 free( T ); 22 } 23 return NULL; 24 } 25 26 /*查找元素在平衡树中的位置*/ 27 Position 28 Find( ElementType X, AvlTree T ) 29 { 30 if( T == NULL ) 31 return NULL; 32 if( X < T->Element ) 33 return Find( X, T->Left ); 34 else 35 if( X > T->Element ) 36 return Find( X, T->Right ); 37 else 38 return T; 39 } 40 41 /*找平衡树的最小元素，递归实现*/ 42 Position 43 FindMin( AvlTree T ) 44 { 45 if( T == NULL ) 46 return NULL; 47 else 48 if( T->Left == NULL ) 49 return T; 50 else 51 return FindMin( T->Left ); 52 } 53 54 /*找平衡树的最大元素，非递归实现*/ 55 Position 56 FindMax( AvlTree T ) 57 { 58 if( T != NULL ) 59 while( T->Right != NULL ) 60 T = T->Right; 61 62 return T; 63 } 64 65 /* START: fig4_36.txt */ 66 67 /*返回节点的高度*/ 68 static int 69 Height( Position P ) 70 { 71 if( P == NULL ) 72 return -1; 73 else 74 return P->Height; 75 } 76 /* END */ 77 78 /*返回两个数的最大值*/ 79 static int 80 Max( int Lhs, int Rhs ) 81 { 82 return Lhs > Rhs ? Lhs : Rhs; 83 } 84 85 /* START: fig4_39.txt */ 86 /* This function can be called only if K2 has a left child */ 87 /* Perform a rotate between a node (K2) and its left child */ 88 /* Update heights, then return new root */ 89 90 /* 91 左左(LL)情况的右旋 92 93 K2 94 / k1 95 K1 => / 96 / k k2 97 k 98 99 */ 100 static Position 101 SingleRotateWithLeft( Position K2 ) 102 { 103 Position K1; 104 105 K1 = K2->Left; 106 K2->Left = K1->Right; 107 K1->Right = K2; 108 109 K2->Height = Max( Height( K2->Left ), Height( K2->Right ) ) + 1; 110 K1->Height = Max( Height( K1->Left ), K2->Height ) + 1; 111 112 return K1; /* New root */ 113 } 114 /* END */ 115 116 /* This function can be called only if K1 has a right child */ 117 /* Perform a rotate between a node (K1) and its right child */ 118 /* Update heights, then return new root */ 119 120 /* 121 右右(RR)情况的左旋 122 123 k1 124 125 k2 => k2 126 / 127 k k1 k 128 129 130 */ 131 static Position 132 SingleRotateWithRight( Position K1 ) 133 { 134 Position K2; 135 136 K2 = K1->Right; 137 K1->Right = K2->Left; 138 K2->Left = K1; 139 140 K1->Height = Max( Height( K1->Left ), Height( K1->Right ) ) + 1; 141 K2->Height = Max( Height( K2->Right ), K1->Height ) + 1; 142 143 return K2; /* New root */ 144 } 145 146 /* START: fig4_41.txt */ 147 /* This function can be called only if K3 has a left */ 148 /* child and K3's left child has a right child */ 149 /* Do the left-right double rotation */ 150 /* Update heights, then return new root */ 151 152 /* 153 154 左右(LR)情况的左右旋转 155 156 k3 k3 157 / / 158 k1 => k2 => k2 159 / / 160 k2 k1 k1 k3 161 162 */ 163 164 static Position 165 DoubleRotateWithLeft( Position K3 ) 166 { 167 /* Rotate between K1 and K2 */ 168 K3->Left = SingleRotateWithRight( K3->Left ); 169 170 /* Rotate between K3 and K2 */ 171 return SingleRotateWithLeft( K3 ); 172 } 173 /* END */ 174 175 /* This function can be called only if K1 has a right */ 176 /* child and K1's right child has a left child */ 177 /* Do the right-left double rotation */ 178 /* Update heights, then return new root */ 179 180 /* 181 182 右左(RL)情况的右左旋转 183 184 k1 k1 185 186 k3 => k2 => k2 187 / / 188 k2 k3 k1 k3 189 190 */ 191 192 static Position 193 DoubleRotateWithRight( Position K1 ) 194 { 195 /* Rotate between K3 and K2 */ 196 K1->Right = SingleRotateWithLeft( K1->Right ); 197 198 /* Rotate between K1 and K2 */ 199 return SingleRotateWithRight( K1 ); 200 } 201 202 203 /* START: fig4_37.txt */ 204 /*平衡树的插入*/ 205 AvlTree 206 Insert( ElementType X, AvlTree T ) 207 { 208 if( T == NULL ) 209 { 210 /* Create and return a one-node tree */ 211 T = malloc( sizeof( struct AvlNode ) ); 212 if( T == NULL ) 213 FatalError( "Out of space!!!" ); 214 else 215 { 216 T->Element = X; T->Height = 0; 217 T->Left = T->Right = NULL; 218 } 219 } 220 else 221 if( X < T->Element ) 222 { 223 T->Left = Insert( X, T->Left ); 224 if( Height( T->Left ) - Height( T->Right ) == 2 ) 225 if( X < T->Left->Element ) 226 T = SingleRotateWithLeft( T ); 227 else 228 T = DoubleRotateWithLeft( T ); 229 } 230 else 231 if( X > T->Element ) 232 { 233 T->Right = Insert( X, T->Right ); 234 if( Height( T->Right ) - Height( T->Left ) == 2 ) 235 if( X > T->Right->Element ) 236 T = SingleRotateWithRight( T ); 237 else 238 T = DoubleRotateWithRight( T ); 239 } 240 /* Else X is in the tree already; we'll do nothing */ 241 242 T->Height = Max( Height( T->Left ), Height( T->Right ) ) + 1; 243 return T; 244 } 245 /* END */ 246 247 AvlTree 248 Delete( ElementType X, AvlTree T ) 249 { 250 printf( "Sorry; Delete is unimplemented; %d remains ", X ); 251 return T; 252 } 253 254 /*返回某节点的值*/ 255 ElementType 256 Retrieve( Position P ) 257 { 258 return P->Element; 259 } 260 261 262 //以下是补充 263 // 264 void PrintTree(AvlTree T) 265 { 266 if (T != NULL) 267 { 268 PrintTree(T->Left); 269 printf("%3d", Retrieve(T)); 270 PrintTree(T->Right); 271 } 272 } 273 274 AvlTree Remove(ElementType X, AvlTree T) 275 { 276 if(T == NULL) 277 return NULL;//没有找到要删除的值,do nothing 278 if(X < T->Element) 279 { 280 T->Left = Remove(X, T->Left); 281 if(Height(T->Right)-Height(T->Left)==2) 282 { 283 //右子树比左子树高2,那么在删除操作之前右子树比左子树高1, 284 //也就是说T的右子树必然不为空,甚至必然有非空子树(高度至少为1). 285 AvlTree s = T->Right; 286 if(Height(s->Left)>Height(s->Right)) 287 T = DoubleRotateWithRight(T);//右双旋转 288 else 289 T = SingleRotateWithRight(T);//右单旋转 290 } 291 else 292 //不需要调整就满足平衡条件的话,只需要重新计算其高度就好 293 T->Height=Max(Height(T->Left),Height(T->Right))+1; 294 } 295 else if(X>T->Element) 296 { 297 298 T->Right = Remove(X,T->Right); 299 if(Height(T->Left)-Height(T->Right)==2) 300 { 301 AvlTree s=T->Left; 302 if(Height(s->Right)>Height(s->Left)) 303 T = DoubleRotateWithLeft(T);//左双旋转 304 else 305 T = SingleRotateWithLeft(T);//左单旋转 306 } 307 else 308 T->Height=Max(Height(T->Left),Height(T->Right))+1; 309 } 310 else 311 { 312 if(T->Left&&T->Right) //T的左右子树都非空,把Remove操作转移到只有一个非空子树的结点或者叶子结点上去 313 { 314 if( Height(T->Left) > Height(T->Right) ) 315 //把Remove操作往更高的那颗子树上转移 316 { 317 //左子树中的最大值 318 T->Element=FindMax(T->Left)->Element; 319 T->Left = Remove(T->Element,T->Left); 320 } 321 else 322 { 323 //右子树中的最小值 324 T->Element=FindMin(T->Right)->Element; 325 T->Right = Remove(T->Element,T->Right); 326 } 327 } 328 else 329 { 330 AvlTree oldnode = T; 331 T=T->Left?T->Left:T->Right; 332 free(oldnode); 333 } 334 } 335 return T; 336 }
参考资料：
数据结构&&AVL树原理、插入操作详解及实现