二叉排序樹

標籤：二叉排序樹

二叉排序樹（Binary Sort Tree）：或者是一顆空樹，或者是具有以下性質的樹：（1）若它的左子樹不空，則左子樹上所以結點的值均小於它的根節點的值；（2）若它的右子樹不空，則右子樹上的所以結點的值均大於它的根節點的值；（3）它的左、右子樹也分別是二叉排序樹。

二叉排序樹的基本操作均可以在O(h)時間內完成（演算法導論p165）。

相關作業碼如下：

int InsertBST(BiTree &T, int key)//遞迴插入{if (T == NULL){T = new BiNode;T->data = key;T->lchild = T->rchild = NULL;return 1;}else{if (key == T->data)return 0;else if (key < T->data)return InsertBST(T->lchild, key);elsereturn InsertBST(T->rchild, key);}}int InsertBST_(BiTree &T, int key)//迭代插入{//find the insert positionBiNode *f = T, *p = T;while (p != NULL){if (key == p->data)return 0;f = p;//記錄上一次訪問的結點p = key < p->data ? p->lchild : p->rchild;}//分配新結點BiNode *q = new BiNode;q->lchild = q->rchild = NULL;q->data = key;//插入if (T == NULL)//若根為空白{T = q;return 1;}if (key < f->data)f->lchild = q;elsef->rchild = q;return 1;}BiNode *SearchBST(BiTree T, int key)//遞迴搜尋{if (!T)return NULL;else{if (key == T->data)return T;else if (key < T->data)return SearchBST(T->lchild, key);elsereturn SearchBST(T->rchild, key);}}BiNode* SearchBST_(BiTree T, int key)//迭代搜尋{while (T != NULL){if (key == T->data)break;T = key < T->data ? T->lchild : T->rchild;}return T;}int DeleteNode(BiNode *&p){BiNode *q;//從二叉排序樹中刪除結點p，並重接它的的左或右子樹if (p == NULL)return 0;if (p->lchild == NULL) //左子樹空則只需重接右子樹{q = p; p = p->rchild; delete q;}else if (p->rchild == NULL)  //右子樹空則只需重接左子樹{q = p; p = p->lchild; delete q;}else//左右子樹均不空{BiNode *s;#if 0//用p的直接前驅代替p，然後刪除p的直接前驅q = p; s = p->lchild;//轉左，然後向右到盡頭while (s->rchild){q = s; s = s->rchild;}p->data = s->data;//s指向被刪除結點,q指向被刪除結點的前驅if (q != p)q->rchild = s->lchild;//重接q的右子樹elseq->lchild = s->lchild;//重接q的左子樹#else//用p的直接後繼代替p，然後刪除p的直接後繼q = p; s = p->rchild;while (s->lchild){q = s; s = s->lchild;}p->data = s->data;if (p != p)q->lchild = s->rchild;elseq->rchild = s->rchild;#endifdelete s;}return 1;}int DeleteBST(BiTree &T, int key){if (T == NULL)return 0;else{if (key == T->data)return DeleteNode(T);else if (key < T->data)return DeleteBST(T->lchild, key);elsereturn DeleteBST(T->rchild, key);}}void CreateBST(BiTree &T, int a[], int n){T = NULL;for (int i = 0; i < n; i++){InsertBST_(T, a[i]);}}void DestoryBST(BiTree &T){if (T == NULL)return;DestoryBST(T->lchild);DestoryBST(T->rchild);delete T; T = NULL;}void InOrderTraverse(BiTree T)//=O(n)時間複雜度{if (T == NULL)return;InOrderTraverse(T->lchild);cout << T->data << " ";InOrderTraverse(T->rchild);}

測試代碼：

int main(){const int n = 10;int a[n] = {3, 2, 8, 6, 1, 4, 5, 7, 1, 3};BiTree T;CreateBST(T, a, n);InOrderTraverse(T);cout << endl;BiNode *p;for (int i = 1; i < 10; i++){p = SearchBST_(T, i);if (p != NULL)cout << p->data << endl;}int b[5]={0, 2, 6, 1, 7};int ret;for (int i = 0; i < 5; i++){ret = DeleteBST(T, b[i]);if (ret == 0)cout << "刪除 " << b[i] << " 失敗" << endl;else{cout << "刪除 " << b[i] << " 後: " ;InOrderTraverse(T);cout << endl;}}DestoryBST(T);getchar();return 0;}

參考：資料結構C語言版、演算法導論（關於刪除結點操作，該書p173頁有註記）