AVL树(Adelson-Velskii-Landis tree)

最后更新于:2022-04-01 16:17:24

AVL树是一个“加上了额外平衡条件”的二叉搜索树。其平衡条件的建立时为了确保树的深度为O(longN)。AVL树要求任何节点左右子树的高度相差不超过1。 插入操作:左-左插入和右-右插入需要单旋转;                  左-右插入和右-左插入需要双旋转。 其实,AVL树的插入删除操作和[二叉查找树的操作](http://blog.csdn.net/u013074465/article/details/41699891)类似,只是需要注意调整树的平衡。 # AVL树 C语言实现 删除操作的旋转示意图: ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/2016-06-07_575683a470879.jpg) ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/2016-06-07_575683a484c5a.jpg) ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/2016-06-07_575683a49bebf.jpg) ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/2016-06-07_575683a4b0e29.jpg) 在下面的代码中,插入删除过程中,对树平衡的调整完全可以封装为函数,显得简洁,但懒人我没写成函数。 ~~~ //3avl_tree.c #include "3avl_tree.h" #include "test.h" AvlTree MakeEmptyAvlTree(AvlTree avltree) { if (avltree != NULL) { MakeEmptyAvlTree(avltree->left); MakeEmptyAvlTree(avltree->right); free(avltree); } return NULL; } Position FindAvlTree(ElementType x, AvlTree avltree) { if (avltree == NULL) return NULL; if (x < avltree->element) return FindAvlTree(x, avltree->left); else if (x > avltree->element) return FindAvlTree(x, avltree->right); else return avltree; } Position FindMinFromAvlTree(AvlTree avltree) { if (avltree == NULL) return NULL; else if (avltree->left == NULL) return avltree; else return FindMinFromAvlTree(avltree->left); } Position FindMaxFromAvlTree(AvlTree avltree) { if (avltree == NULL) return NULL; else if (avltree->right == NULL) return avltree; else return FindMaxFromAvlTree(avltree->right); } static int HeightAvlTree(AvlTree avltree) { if (avltree == NULL) return -1; else return avltree->height; } static int Max(int lhs, int rhs) { return lhs > rhs ? lhs : rhs; } //左-左插入,使得k2失去平衡,因此需要在k2处单旋转 static Position SingleRotateWithLeft(Position k2) { Position k1; k1 = k2->left; k2->left = k1->right; k1->right = k2; k2->height = Max(HeightAvlTree(k2->left), HeightAvlTree(k2->right)) + 1; k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1; return k1; } //右-右插入,使得k1失去平衡,因此需要在k1处单旋转 static Position SigleRotateWithRight(Position k1) { Position k2; k2 = k1->right; k1->right = k2->left; k2->left = k1; k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1; k2->height = Max(HeightAvlTree(k2->left), HeightAvlTree(k2->right)) + 1; return k2; } //左-右插入,使得k3失去平衡,因此在k3处双旋转 //双旋转其实是两次单旋转操作 //先对k3的左子树进行一次右-右单旋转操作 //然后对k3进行一次左-左单旋转操作 static Position DoubleRotateWithLeft1(Position k3) { //双旋转,递归 k3->left = SigleRotateWithRight(k3->left); return SingleRotateWithLeft(k3); } static Position DoubleRotateWithLeft2(Position k3) { Position k1, k2; k1 = k3->left; k2 = k1->right; k1->right = k2->left; k2->left = k1; k3->left = k2->right; k2->right = k3; k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1; k3->height = Max(HeightAvlTree(k3->left), HeightAvlTree(k3->right)) + 1; k2->height = Max(k1->height, k3->height) + 1; return k2; } //右-左插入,使得k1失去平衡,因此在k1处双旋转 //双旋转其实是两次单旋转操作 //先对k1的右子树进行一次左-左单旋转操作 //然后对k1进行一次右-右单旋转操作 static Position DoubleRotateWithRight(Position k1) { k1->right = SingleRotateWithLeft(k1->right); return SigleRotateWithRight(k1); } AvlTree InsertToAvlTree(ElementType x, AvlTree avltree) { if (avltree == NULL) { //如果树为空,分配空间,插入节点 avltree = (struct AvlNode*)malloc(sizeof(struct AvlNode)); if (avltree == NULL) { cout << "malloc failed." << endl; } else { avltree->element = x; avltree->left = avltree->right = NULL; } } //插入值小于根节点的值,在左子树插入节点 else if (x < avltree->element) { avltree->left = InsertToAvlTree(x, avltree->left); //树失去了平衡,对树进行调整,这几个调整平衡的语句可以封装下 if (HeightAvlTree(avltree->left) - HeightAvlTree(avltree->right) == 2) { if (x < avltree->left->element) //左-左,进行左单旋转 avltree = SingleRotateWithLeft(avltree); else //左-右,双旋转 avltree = DoubleRotateWithLeft1(avltree); } } //插入值大于根节点的值,在右子树插入节点 else if (x > avltree->element) { avltree->right = InsertToAvlTree(x, avltree->right); //树失去了平衡,对树进行调整,这几个调整平衡的语句可以封装下 if (HeightAvlTree(avltree->right) - HeightAvlTree(avltree->left) == 2) { if (x > avltree->right->element) avltree = SigleRotateWithRight(avltree); //右-右,单旋转 else //右-左,双旋转 avltree = DoubleRotateWithRight(avltree); } } avltree->height = Max(HeightAvlTree(avltree->left), HeightAvlTree(avltree->right)) + 1; return avltree; } //删除操作类似于二叉树的操作,只是需要将树调整平衡 AvlTree DeleteFromAvlTree(ElementType x, AvlTree avltree) { Position temp; if (avltree == NULL) return NULL; //在左子树寻找删除位置,删除后可能导致左子树高度变小,可能要调整树 if (x < avltree->element) { avltree->left = DeleteFromAvlTree(x, avltree->left); //如果右子树高度比左子树高度超过1,那么需要将树调整平衡,这几个调整平衡的语句可以封装下 if (2 == (HeightAvlTree(avltree->right) - HeightAvlTree(avltree->left))) { //如果是右-左型,则要进行左单旋、右单旋;即进行如下双旋操作 //否则为右-右型,需要进行左单旋操作 if (HeightAvlTree(avltree->right->left) > HeightAvlTree(avltree->right->right)) avltree = DoubleRotateWithRight(avltree); else avltree = SigleRotateWithRight(avltree); } } //在右子树寻找删除位置,删除后可能导致右子树高度变小,可能要调整树 else if (x > avltree->element) { avltree->right = DeleteFromAvlTree(x, avltree->right); //这几个调整平衡的语句可以封装下 if (2 == (HeightAvlTree(avltree->left) - HeightAvlTree(avltree->right))) { if (HeightAvlTree(avltree->left->right) > HeightAvlTree(avltree->left->left)) avltree = DoubleRotateWithLeft1(avltree); else avltree = SingleRotateWithLeft(avltree); } } //找到要删除的元素,该节点有两个孩子 //那么将节点的值用右子树中最小的值A替换,然后在右子树中删除A else if (avltree->left && avltree->right){ avltree->element = (FindMinFromAvlTree(avltree->right))->element; avltree->right = DeleteFromAvlTree(avltree->element, avltree->right); } //找到要删除的节点,该节点为叶子节点或仅有一个孩子 else { temp = avltree; if (avltree->left == NULL) //当节点只有右孩子或为叶节点时 avltree = avltree->right; else if (avltree->right == NULL) //当节点只有左孩子时 avltree = avltree->left; free(temp); } if (avltree) avltree->height = Max(HeightAvlTree(avltree->right), HeightAvlTree(avltree->left)) + 1; return avltree; } void VisitElement(AvlTree tree) { cout << tree->element << " "; } //先序遍历树 void PreOrderTraversal(AvlTree tree) { if (tree == NULL) return; VisitElement(tree); PreOrderTraversal(tree->left); PreOrderTraversal(tree->right); } //中序遍历树 void InOrderTraversal(AvlTree tree) { if (tree == NULL) return; InOrderTraversal(tree->left); VisitElement(tree); InOrderTraversal(tree->right); } int main() { AvlTree tree = NULL; tree = MakeEmptyAvlTree(tree); tree = InsertToAvlTree(1, tree); tree = InsertToAvlTree(3, tree); tree = InsertToAvlTree(5, tree); tree = InsertToAvlTree(122, tree); tree = InsertToAvlTree(7, tree); tree = InsertToAvlTree(9, tree); tree = InsertToAvlTree(11, tree); tree = InsertToAvlTree(21, tree); Position min = FindMinFromAvlTree(tree);; cout << "删除前,树高:" << HeightAvlTree(tree) << endl; cout << endl << "删除前,先序遍历:" << endl; PreOrderTraversal(tree); cout << endl << "删除前,中序遍历:" << endl; InOrderTraversal(tree); DeleteFromAvlTree(7, tree); cout << endl; cout << "删除后,树高:" << HeightAvlTree(tree) << endl; cout << endl << "删除后,先序遍历:" << endl; PreOrderTraversal(tree); cout << endl << "删除后,中序遍历:" << endl; InOrderTraversal(tree); } ~~~ ~~~ //3avl_tree.h typedef int ElementType; #ifndef TEST_AVL_TREE_H #define TEST_AVL_TREE_H struct AvlNode; typedef struct AvlNode *Position; typedef struct AvlNode *AvlTree; struct AvlNode { ElementType element; AvlTree left; AvlTree right; int height; }; AvlTree MakeEmptyAvlTree(AvlTree avltree); Position FindAvlTree(ElementType x, AvlTree avltree); Position FindMinFromAvlTree(AvlTree avltree); Position FindMaxFromAvlTree(AvlTree avltree); AvlTree InsertToAvlTree(ElementType x, AvlTree avltree); AvlTree DeleteFromAvlTree(ElementType x, AvlTree avltree); ElementType RetrieveAvlTree(Position pos); #endif ~~~ 程序执行结果: ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/2016-06-07_575683a4cb038.jpg) # [Avl的C++实现](http://blog.csdn.net/u013074465/article/details/44748497)
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