【數據結構】二叉排序樹

二叉排序樹（Binary Sort Tree）又稱二叉查找樹（Binary Search Tree），亦稱二叉搜索樹。

特點

二叉排序樹或者是一棵空樹，或者是具有下列性質的二叉樹：

1、若左子樹不空，則左子樹上所有結點的值均小于它的根結點的值；

2、若右子樹不空，則右子樹上所有結點的值均大于它的根結點的值；

3、左、右子樹也分別為二叉排序樹；

4、沒有鍵值相等的節點。

特性

二叉排序樹通常采用二叉鏈表作為存儲結構。中序遍歷二叉排序樹可得到一個依據關鍵字的有序序列，一個無序序列可以通過構造一棵二叉排序樹變成一個有序序列，構造樹的過程即是對無序序列進行排序的過程。每次插入的新的結點都是二叉排序樹上新的葉子結點，在進行插入操作時，不必移動其它結點，只需改動某個結點的指針，由空變為非空即可。搜索、插入、刪除的時間復雜度等于樹高，期望O(logn)，最壞O(n)（數列有序，樹退化成線性表，如右斜樹）。

查找算法

步驟：

1、若子樹為空，查找不成功。

2、二叉樹若根結點的關鍵字值等于查找的關鍵字，成功。

3、否則，若小于根結點的關鍵字值，遞歸查左子樹。

4、若大于根結點的關鍵字值，遞歸查右子樹。

插入算法

步驟：

1、執行查找算法，找出被插結點的父親結點。

2、判斷被插結點是其父親結點的左、右兒子。將被插結點作為葉子結點插入。

3、若二叉樹為空。則首先單獨生成根結點。

刪除算法

步驟：

1.若*p結點為葉子結點，即PL(左子樹)和PR(右子樹)均為空樹。由于刪去葉子結點不破壞整棵樹的結構，則只需修改其雙親結點的指針即可。

?2.若*p結點只有左子樹PL或右子樹PR，此時只要令PL或PR直接成為其雙親結點*f的左子樹（當*p是左子樹）或右子樹（當*p是右子樹）即可，作此修改也不破壞二叉排序樹的特性。

?3.若*p結點的左子樹和右子樹均不空。在刪去*p之后，為保持其它元素之間的相對位置不變，可按中序遍歷保持有序進行調整。比較好的做法是，找到*p的直接前驅（或直接后繼）*s，用*s來替換結點*p，然后再刪除結點*s。

二叉排序樹的實現練習（Java）

public class BinarySortTree {private TreeNode root=null;/*** 獲取樹的高度* @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);}}/*** 獲取樹的節點數* @param subTree* @return*/private int size(TreeNode subTree){if(subTree == null){return 0;}else{return size(subTree.leftChild)+size(subTree.rightChild)+1;}}/*** 前序遍歷  * @param subTree*/public void preOrder(TreeNode subTree){  if(subTree!=null){visted(subTree);  preOrder(subTree.leftChild); preOrder(subTree.rightChild); }}  /*** 中序遍歷  * @param subTree*/public void inOrder(TreeNode subTree){  if(subTree!=null){  inOrder(subTree.leftChild);  visted(subTree);  inOrder(subTree.rightChild);  }  }  /*** 后續遍歷  * @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.");}}/*** 查詢* @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;}}/*** 刪除節點* @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){/*如果左右子樹為空，怎直接刪除該節點*/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){/*如果左子樹不為空而右子樹為空，則直接用左子樹根節點替換刪除節點*/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){/*如果左子樹為空而右子樹不為空，則直接用右子樹根節點替換刪除節點*/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{/*左右子樹都不為空的情況下,直接找前驅替代P，并釋放*/TreeNode p = subTree.leftChild;if(p.rightChild == null){/*P是刪除節點的左子樹最大值，即前驅，替換刪除節點*/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是刪除節點的左子樹最大值(即前驅)，替換刪除節點*/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;}}}/** * 二叉樹的節點數據結構 */  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;}}