3.平衡二叉树
平衡二叉树,又称AVL树,它是一种特殊的二叉排序树。
3.1 平衡二叉树的四种自旋
这个左旋、右旋,在方向上和我观念里的是相反的。
查了之后才知道:
1、外侧插入:LL、RR,都是在最边边上。
2、内侧插入:LR、RL,往里面来了些。
(1)LL旋转和RR旋转:
void RR_Rotate(AVLTree *root){ AVLTreeNode* rchild = (*root)->Right; (*root)->Right = rchild->Left; rchild->Left = *root; *root = rchild; } void LL_Rotate(AVLTree *root) { AVLTreeNode* lchild = (*root)->Left; (*root)->Left = lchild->Right; lchild->Right = *root; *root = lchild; }
(2)LR,RL型旋转(插入的节点在CL上 还是CR上是没有影响的)
LR型旋转图解
void LR_Rotate(AVLTree *root) { RR_Rotate(&(*root)->Left); return LL_Rotate(root); } void RL_Rotate(AVLTree *root) { LL_Rotate(&(*root)->Right); RR_Rotate(root); }
小结:几个子树在水平位置上顺序是不会变的!根节点和根节点的子树在水平上的逻辑位置也是不会变的。
3.2 平衡二叉树的插入
插入算法就是出现不平衡状态时,判断需要使用哪种旋转方式来使得二叉树保持平衡:
AVLTree InsertAVLTree(AVLTree root, int x) { if (root == NULL) { root = new AVLTreeNode; root->Left = NULL; root->Right = NULL; root->data = x; return root; } if (x > root->data) { root->Right = InsertAVLTree(root->Right, x); //递归返回插入位置的父节点或者祖父……,如果失去了平衡 if (height(root->Left) - height(root->Right) == -2) { //如果插入的值大于,当前节点的左孩子节点,说明该节点是插在root的右子树上的 if (x > root->Left->data) RR_Rotate(&root); else RL_Rotate(&root); } } else if (x < root->data) { root->Left = InsertAVLTree(root->Left, x); if (height(root->Left) - height(root->Right) == 2) { if (x < root->Left->data) LL_Rotate(&root); else LR_Rotate(&root); } } else { cout << "the number is already included." << endl; return NULL; } return root; }
3.3 平衡二叉树的删除
之前写过二叉排序树的节点的删除的话,这里会好写很多,就是多出来一个判断从哪个子树删除节点的问题。
void AVLTreeDel(AVLTree *root, int data) { if (!*root) { cout << "delete failed" << endl; return; } AVLTreeNode *p = *root; if (data == p->data) { //左右子树都非空 if (p->Left && p->Right) { //在高度更大的那个子树上进行删除操作 //进左子树,右转到底,进右子树,左转到底,转弯碰壁,杀孩子。 if (height(p->Left) > height(p->Right)) { AVLTreeNode *pre=NULL,*q = p->Left; if (!q->Right) q->Right = p->Right; else { while (q->Right) { pre = q; q = q->Right; } pre->Right = q->Left; q->Left = p->Left; q->Right = p->Right; } *root = q; } else { AVLTreeNode *pre = NULL, *q = p->Right; if (!q->Left) q->Left = p->Left; else { while (q->Left) { pre = q; q = q->Left; } pre->Left = q->Right; q->Left = p->Left; q->Right = p->Right; } *root=q; } } else (*root) = (*root)->Left ? (*root)->Left : (*root)->Right; delete p; } else if (data < p->data){//要删除的节点在左子树中 //在左子树中进行递归删除 AVLTreeDel(&(*root)->Left, data); //判断是否仍然满足平衡条件 if (height(p->Right) - height(p->Left) == 2){ //如果当前节点右孩子的左子树更高 if (height(p->Right->Left) > height(p->Right->Right)) RL_Rotate(root); else RR_Rotate(root); } } else{ AVLTreeDel(&(*root)->Right, data); if (height(p->Left) - height(p->Right) == 2) { if (height((*root)->Left->Left) > height((*root)->Left->Right)) LL_Rotate(root); else LR_Rotate(root); } } }
https://www.2cto.com/kf/201702/556250.html
完整代码(2019.1.20):
#pragma once #include "main.h" typedef struct AVLTreeNode { int data; struct AVLTreeNode *Left; struct AVLTreeNode *Right; }*AVLTree; int height(AVLTree L) { if (L == NULL) return 0; int left = height(L->Left); int right = height(L->Right); return left >= right ? left + 1 : right + 1; } void RR_Rotate(AVLTree *root){ AVLTreeNode* Right = (*root)->Right; (*root)->Right = Right->Left; Right->Left = *root; *root = Right; } void LL_Rotate(AVLTree *root) { AVLTreeNode* Left = (*root)->Left; (*root)->Left = Left->Right; Left->Right = *root; *root = Left; } void LR_Rotate(AVLTree *root) { RR_Rotate(&(*root)->Left); return LL_Rotate(root); } void RL_Rotate(AVLTree *root) { LL_Rotate(&(*root)->Right); RR_Rotate(root); } AVLTree AVLTreeInsert(AVLTree root, int x) { if (root == NULL) { root = new AVLTreeNode; root->Left = NULL; root->Right = NULL; root->data = x; return root; } if (x > root->data) { root->Right = AVLTreeInsert(root->Right, x); //递归返回插入位置的父节点或者祖父……,如果失去了平衡 if (height(root->Left) - height(root->Right) == -2) { //如果插入的值大于,当前节点的右孩子节点,说明该节点是插在root的右子树上的 //if (x > root->Left->data) RR_Rotate(&root);不能保证该节点一定有左子树 if (x > root->Right->data)RR_Rotate(&root); else RL_Rotate(&root); } } else if (x < root->data) { root->Left = AVLTreeInsert(root->Left, x); if (height(root->Left) - height(root->Right) == 2) { if (x < root->Left->data) LL_Rotate(&root); else LR_Rotate(&root); } } else { cout << "the number is already included." << endl; return NULL; } return root; } AVLTree AVLTreeCreat(int *a, int length) { AVLTree T = NULL; for (int i = 0; i < length; i++) { T = AVLTreeInsert(T, a[i]); } return T; } AVLTreeNode* AVLFind(AVLTree T, int x) { AVLTreeNode *p = T; while (p) { if (x == p->data) break; p = x > p->data ? p->Right : p->Left; } return p; } AVLTree AVLMax(AVLTree p) { if (!p) return NULL; if (p->Right == NULL) return p; return AVLMax(p->Right); } AVLTree AVLMin(AVLTree p) { if (!p) return NULL; if (p->Left == NULL) return p; return AVLMin(p->Left); } void AVLTreeDel(AVLTree *root, int data) { if (!*root) { cout << "delete failed" << endl; return; } AVLTreeNode *p = *root; if (data == p->data) { //左右子树都非空 if (p->Left && p->Right) { //在高度更大的那个子树上进行删除操作 //进左子树,右转到底,进右子树,左转到底,转弯碰壁,杀孩子。 if (height(p->Left) > height(p->Right)) { AVLTreeNode *pre=NULL,*q = p->Left; if (!q->Right) q->Right = p->Right; else { while (q->Right) { pre = q; q = q->Right; } pre->Right = q->Left; q->Left = p->Left; q->Right = p->Right; } *root = q; } else { AVLTreeNode *pre = NULL, *q = p->Right; if (!q->Left) q->Left = p->Left; else { while (q->Left) { pre = q; q = q->Left; } pre->Left = q->Right; q->Left = p->Left; q->Right = p->Right; } *root=q; } } else (*root) = (*root)->Left ? (*root)->Left : (*root)->Right; delete p; } else if (data < p->data){//要删除的节点在左子树中 //在左子树中进行递归删除 AVLTreeDel(&(*root)->Left, data); //判断是否仍然满足平衡条件 if (height(p->Right) - height(p->Left) == 2){ //如果当前节点右孩子的左子树更高 if (height(p->Right->Left) > height(p->Right->Right)) RL_Rotate(root); else RR_Rotate(root); } } else{ AVLTreeDel(&(*root)->Right, data); if (height(p->Left) - height(p->Right) == 2) { if (height((*root)->Left->Left) > height((*root)->Left->Right)) LL_Rotate(root); else LR_Rotate(root); } } } void Preorder(AVLTree T) { if (!T)return; cout << T->data << " "; Preorder(T->Left); Preorder(T->Right); } void Inorder(AVLTree T) { if (!T)return; Inorder(T->Left); cout << T->data << " "; Inorder(T->Right); } void Postorder(AVLTree T) { if (!T)return; Postorder(T->Left); Postorder(T->Right); cout << T->data << " "; } void checkCreat() { int length = 10; int *a = getNoRepateRandomArray(length, 10); for (int i = 0; i < length; i++) { cout << a[i] << ","; } cout << endl; AVLTree T = AVLTreeCreat(a, length); int t = rand() % length; AVLTreeDel(&T, a[t]); for (int i = t; i < length - 1; i++) { a[i] = a[i + 1]; } Preorder(T); cout << endl; Inorder(T); cout << endl; Postorder(T); cout << endl; free(a); }