【数据结构】二叉排序树

二叉排序树（Binary Sort Tree）又称二叉查找树（Binary Search Tree），亦称二叉搜索树。

特点

二叉排序树或者是一棵空树，或者是具有下列性质的二叉树：

1、若左子树不空，则左子树上所有结点的值均小于它的根结点的值；

2、若右子树不空，则右子树上所有结点的值均大于它的根结点的值；

3、左、右子树也分别为二叉排序树；

4、没有键值相等的节点。

特性

二叉排序树通常采用二叉链表作为存储结构。中序遍历二叉排序树可得到一个依据关键字的有序序列，一个无序序列可以通过构造一棵二叉排序树变成一个有序序列，构造树的过程即是对无序序列进行排序的过程。每次插入的新的结点都是二叉排序树上新的叶子结点，在进行插入操作时，不必移动其它结点，只需改动某个结点的指针，由空变为非空即可。搜索、插入、删除的时间复杂度等于树高，期望O(logn)，最坏O(n)（数列有序，树退化成线性表，如右斜树）。

查找算法

步骤：

1、若子树为空，查找不成功。

2、二叉树若根结点的关键字值等于查找的关键字，成功。

3、否则，若小于根结点的关键字值，递归查左子树。

4、若大于根结点的关键字值，递归查右子树。

插入算法

步骤：

1、执行查找算法，找出被插结点的父亲结点。

2、判断被插结点是其父亲结点的左、右儿子。将被插结点作为叶子结点插入。

3、若二叉树为空。则首先单独生成根结点。

删除算法

步骤：

1.若*p结点为叶子结点，即PL(左子树)和PR(右子树)均为空树。由于删去叶子结点不破坏整棵树的结构，则只需修改其双亲结点的指针即可。

2.若*p结点只有左子树PL或右子树PR，此时只要令PL或PR直接成为其双亲结点*f的左子树（当*p是左子树）或右子树（当*p是右子树）即可，作此修改也不破坏二叉排序树的特性。

3.若*p结点的左子树和右子树均不空。在删去*p之后，为保持其它元素之间的相对位置不变，可按中序遍历保持有序进行调整。比较好的做法是，找到*p的直接前驱（或直接后继）*s，用*s来替换结点*p，然后再删除结点*s。

二叉排序树的实现练习（Java）

public class BinarySortTree {private TreeNode root=null;/*** 获取树的高度* @param subTree * @return*/private int height(TreeNode subTree){if(subTree == null){return 0;}else{int i = height(subTree.leftChild);int j = height(subTree.rightChild);return (i>j)?(i+1):(j+1);}}/*** 获取树的节点数* @param subTree* @return*/private int size(TreeNode subTree){if(subTree == null){return 0;}else{return size(subTree.leftChild)+size(subTree.rightChild)+1;}}/*** 前序遍历  * @param subTree*/public void preOrder(TreeNode subTree){  if(subTree!=null){visted(subTree);  preOrder(subTree.leftChild); preOrder(subTree.rightChild); }}  /*** 中序遍历  * @param subTree*/public void inOrder(TreeNode subTree){  if(subTree!=null){  inOrder(subTree.leftChild);  visted(subTree);  inOrder(subTree.rightChild);  }  }  /*** 后续遍历  * @param subTree*/public void postOrder(TreeNode subTree) {  if (subTree != null) {  postOrder(subTree.leftChild);  postOrder(subTree.rightChild);  visted(subTree);  }  }public void visted(TreeNode subTree){  System.out.print(subTree.data+",");}/*** 插入* @param subTree* @param iv*/public void insertNote(TreeNode subTree, int iv){TreeNode newNode = new TreeNode(iv);if(subTree == null){this.root = newNode;}else if(subTree.data > iv){if(subTree.leftChild == null){subTree.leftChild = newNode;newNode.parent = subTree;}else{insertNote(subTree.leftChild, iv);}}else if(subTree.data < iv){if(subTree.rightChild == null){subTree.rightChild = newNode;newNode.parent = subTree;}else{insertNote(subTree.rightChild, iv);}}else{System.out.println("node has exist.");}}/*** 查询* @param subTree* @param fv* @return*/public boolean findNote(TreeNode subTree, int fv){if(subTree == null){return false;}else if(subTree.data > fv){return findNote(subTree.leftChild, fv);}else if(subTree.data < fv){return findNote(subTree.rightChild, fv);}else{return true;}}/*** 删除节点* @param subTree* @param iv*/public void deleteNote(TreeNode subTree, int dv){if(subTree == null){System.out.println("BST is empty.");}else if(subTree.data > dv){deleteNote(subTree.leftChild, dv);}else if(subTree.data < dv){deleteNote(subTree.rightChild, dv);}else{if(subTree.leftChild == null && subTree.rightChild == null){/*如果左右子树为空，怎直接删除该节点*/if(subTree.parent == null){this.root = null;}else if(subTree.parent.leftChild == subTree){subTree.parent.leftChild = null;subTree.parent = null;}else if(subTree.parent.rightChild == subTree){subTree.parent.rightChild = null;subTree.parent = null;}}else if(subTree.leftChild != null && subTree.rightChild == null){/*如果左子树不为空而右子树为空，则直接用左子树根节点替换删除节点*/if(subTree.parent == null){this.root = subTree.leftChild;}else if(subTree.parent.leftChild == subTree){subTree.parent.leftChild = subTree.leftChild;}else if(subTree.parent.rightChild == subTree){subTree.parent.rightChild = subTree.leftChild;}subTree.leftChild.parent = subTree.parent;subTree.parent = null;subTree.leftChild = null;subTree = null;}else if(subTree.leftChild == null && subTree.rightChild != null){/*如果左子树为空而右子树不为空，则直接用右子树根节点替换删除节点*/if(subTree.parent == null){this.root = subTree.leftChild;}else if(subTree.parent.leftChild == subTree){subTree.parent.leftChild = subTree.rightChild;}else if(subTree.parent.rightChild == subTree){subTree.parent.rightChild = subTree.rightChild;}subTree.rightChild.parent = subTree.parent;subTree.parent = null;subTree.rightChild = null;subTree = null;}else{/*左右子树都不为空的情况下,直接找前驱替代P，并释放*/TreeNode p = subTree.leftChild;if(p.rightChild == null){/*P是删除节点的左子树最大值，即前驱，替换删除节点*/if(subTree.parent == null){this.root = p;}else if(subTree.parent.leftChild == subTree){subTree.parent.leftChild = p;}else if(subTree.parent.rightChild == subTree){subTree.parent.rightChild = p;}p.parent = subTree.parent;p.rightChild = subTree.rightChild;subTree.rightChild.parent = p;}else{while(p.rightChild != null){p = p.rightChild;}if(p.leftChild != null){p.parent.rightChild = p.leftChild;p.leftChild.parent = p.parent;p.parent = null;p.leftChild = null;}else{p.parent.rightChild = null;p.parent = null;}/*P是删除节点的左子树最大值(即前驱)，替换删除节点*/if(subTree.parent == null){this.root = p;}else if(subTree.parent.leftChild == subTree){subTree.parent.leftChild = p;}else if(subTree.parent.rightChild == subTree){subTree.parent.rightChild = p;}p.parent = subTree.parent;p.leftChild = subTree.leftChild;subTree.leftChild.parent = p;p.rightChild = subTree.rightChild;subTree.rightChild.parent = p;}subTree.parent = null;subTree.leftChild = null;subTree.rightChild = null;subTree = null;}}}/** * 二叉树的节点数据结构 */  private class  TreeNode{private int data;private TreeNode parent = null;private TreeNode leftChild=null;private TreeNode rightChild=null;public TreeNode(int data){  this.data=data; }  }public static void main(String[] args) {int[] tns = {1,3,4,6,7,8,10,13,14};for(int dv:tns){BinarySortTree bst = createBST();System.out.println("===============delete "+dv+" demo=====================");System.out.println("findNote("+dv+"):"+bst.findNote(bst.root, dv));    System.out.println("before delete inOrder:");bst.inOrder(bst.root);System.out.println("");bst.deleteNote(bst.root, dv);System.out.println("after delete inOrder:");bst.inOrder(bst.root);System.out.println("");System.out.println("");}}public static BinarySortTree createBST(){BinarySortTree bst = new BinarySortTree();bst.insertNote(bst.root, 8);bst.insertNote(bst.root, 3);bst.insertNote(bst.root, 10);bst.insertNote(bst.root, 1);bst.insertNote(bst.root, 6);bst.insertNote(bst.root, 14);bst.insertNote(bst.root, 4);bst.insertNote(bst.root, 7);bst.insertNote(bst.root, 13);return bst;}}