二叉搜索树(Binary Search Tree)（Java实现）

文章目录

1、二叉搜索树
- 1.1、基本概念
- 1.2、树的节点（BinaryNode）
- 1.3、构造器和成员变量
- 1.3、公共方法（public method）
- 1.4、比较函数
- 1.5、contains 函数
- 1.6、findMin
- 1.7、findMax
- 1.8、insert
- 1.9、remove
二、完整代码实现（Java）

1、二叉搜索树

1.1、基本概念

二叉树的一个性质是一棵平均二叉树的深度要比节点个数N小得多。分析表明其平均深度为O(N)\mathcal{O}(\sqrt{N})O(N),而对于特殊类型的二叉树，即二叉查找树(binary search tree),其深度的平均值为O(logN)\mathcal{O}(log N)O(logN)。

二叉查找树的性质：对于树中的每个节点X，它的左子树中所有项的值小于X中的项，而它的右子树中所有项的值大于X中的项。

由于树的递归定义，通常是递归地编写那些操作的例程。因为二叉查找树的平均深度为O(logN)\mathcal{O}(log N)O(logN)，所以一般不必担心栈空间被用尽。

1.2、树的节点（BinaryNode）

二叉查找树要求所有的项都能够排序，有两种实现方式；

对象实现接口 Comparable, 树中的两项使用compareTo方法进行比较；
使用一个函数对象，在构造器中传入一个比较器；

本篇文章采用了构造器重载，并定义了myCompare方法，使用了泛型，因此两种方式都支持，在后续的代码实现中可以看到。

节点定义：

    /*** 节点** @param <AnyType>*/private static class BinaryNode<AnyType> {BinaryNode(AnyType theElement) {this(theElement, null, null);}BinaryNode(AnyType theElement, BinaryNode<AnyType> left, BinaryNode<AnyType> right) {element = theElement;left = left;right = right;}AnyType element; // the data in the nodeBinaryNode<AnyType> left; // Left childBinaryNode<AnyType> right; // Right child}

1.3、构造器和成员变量

    private BinaryNode<AnyType> root;private Comparator<? super AnyType> cmp;/*** 无参构造器*/public BinarySearchTree() {this(null);}/*** 带参构造器，比较器** @param c 比较器*/public BinarySearchTree(Comparator<? super AnyType> c) {root = null;cmp = c;}

关于比较器的知识可以参考下面这篇文章：
Java中Comparator的使用
关于泛型的知识可以参考下面这篇文章：
如何理解 Java 中的 <T extends Comparable<? super T>>

1.3、公共方法（public method）

主要包括插入，删除，找到最大值、最小值，清空树，查看元素是否包含；

/*** 清空树*/public void makeEmpty() {root = null;}public boolean isEmpty() {return root == null;}public boolean contains(AnyType x){return contains(x,root);}public AnyType findMin(){if (isEmpty()) throw new BufferUnderflowException();return findMin(root).element;}public AnyType findMax(){if (isEmpty()) throw new BufferUnderflowException();return findMax(root).element;}public void insert(AnyType x){root = insert(x, root);}public void remove(AnyType x){root = remove(x,root);}

1.4、比较函数

如果有比较器，就使用比较器，否则要求对象实现了Comparable接口；

    private int myCompare(AnyType lhs, AnyType rhs) {if (cmp != null) {return cmp.compare(lhs, rhs);} else {return lhs.compareTo(rhs);}}

1.5、contains 函数

本质就是一个树的遍历；

    private boolean contains(AnyType x, BinaryNode<AnyType> t) {if (t == null) {return false;}int compareResult = myCompare(x, t.element);if (compareResult < 0) {return contains(x, t.left);} else if (compareResult > 0) {return contains(x, t.right);} else {return true;}}

1.6、findMin

因为二叉搜索树的性质，最小值一定是树的最左节点,要注意树为空的情况。

    /*** Internal method to find the smallest item in a subtree* @param t the node that roots the subtree* @return node containing the smallest item*/private BinaryNode<AnyType> findMin(BinaryNode<AnyType> t) {if (t == null) {return null;}if (t.left == null) {return t;}return findMin(t.left);}

1.7、findMax

最右节点；

    /*** Internal method to find the largest item in a subtree* @param t the node that roots the subtree* @return the node containing the largest item*/private BinaryNode<AnyType> findMax(BinaryNode<AnyType> t){if (t == null){return null;}if (t.right == null){return t;}return findMax(t.right);}

1.8、insert

这个主要是根据二叉搜索树的性质，注意当树为空的情况，就可以加入新的节点了，还有当该值已经存在时，默认不进行操作；

    /*** Internal method to insert into a subtree* @param x the item to insert* @param t the node that roots the subtree* @return the new root of the subtree*/private BinaryNode<AnyType> insert(AnyType x, BinaryNode<AnyType> t){if (t == null){return new BinaryNode<>(x,null,null);}int compareResult = myCompare(x,t.element);if (compareResult < 0){t.left = insert(x,t.left);}else if (compareResult > 0){t.right = insert(x,t.right);}else{//Duplicate; do nothing}return t;}

1.9、remove

注意当空树时，返回null；
最后一个三元表达式，是在之前已经排除掉节点有两个儿子的情况下使用的。

    /*** Internal method to remove from a subtree* @param x the item to remove* @param t the node that roots the subtree* @return the new root of the subtree*/private BinaryNode<AnyType> remove(AnyType x, BinaryNode<AnyType> t){if (t == null){return t; // Item not found ,do nothing}int compareResult = myCompare(x,t.element);if (compareResult < 0){t.left = remove(x,t.left);}else if (compareResult > 0){t.right = remove(x,t.right);}else if (t.left !=null && t.right!=null){//Two childrent.element = findMin(t.right).element;t.right = remove(t.element,t.right);}elset = (t.left !=null) ? t.left:t.right;return t;}

二、完整代码实现（Java）

/*** @author LongRookie* @description: 二叉搜索树* @date 2021/6/26 19:41*/import com.sun.source.tree.BinaryTree;import java.nio.BufferUnderflowException;
import java.util.Comparator;/*** 二叉搜索树*/
public class BinarySearchTree<AnyType extends Comparable<? super AnyType>> {/*** 节点** @param <AnyType>*/private static class BinaryNode<AnyType> {BinaryNode(AnyType theElement) {this(theElement, null, null);}BinaryNode(AnyType theElement, BinaryNode<AnyType> left, BinaryNode<AnyType> right) {element = theElement;left = left;right = right;}AnyType element; // the data in the nodeBinaryNode<AnyType> left; // Left childBinaryNode<AnyType> right; // Right child}private BinaryNode<AnyType> root;private Comparator<? super AnyType> cmp;/*** 无参构造器*/public BinarySearchTree() {this(null);}/*** 带参构造器，比较器** @param c 比较器*/public BinarySearchTree(Comparator<? super AnyType> c) {root = null;cmp = c;}/*** 清空树*/public void makeEmpty() {root = null;}public boolean isEmpty() {return root == null;}public boolean contains(AnyType x){return contains(x,root);}public AnyType findMin(){if (isEmpty()) throw new BufferUnderflowException();return findMin(root).element;}public AnyType findMax(){if (isEmpty()) throw new BufferUnderflowException();return findMax(root).element;}public void insert(AnyType x){root = insert(x, root);}public void remove(AnyType x){root = remove(x,root);}private int myCompare(AnyType lhs, AnyType rhs) {if (cmp != null) {return cmp.compare(lhs, rhs);} else {return lhs.compareTo(rhs);}}private boolean contains(AnyType x, BinaryNode<AnyType> t) {if (t == null) {return false;}int compareResult = myCompare(x, t.element);if (compareResult < 0) {return contains(x, t.left);} else if (compareResult > 0) {return contains(x, t.right);} else {return true;}}/*** Internal method to find the smallest item in a subtree* @param t the node that roots the subtree* @return node containing the smallest item*/private BinaryNode<AnyType> findMin(BinaryNode<AnyType> t) {if (t == null) {return null;}if (t.left == null) {return t;}return findMin(t.left);}/*** Internal method to find the largest item in a subtree* @param t the node that roots the subtree* @return the node containing the largest item*/private BinaryNode<AnyType> findMax(BinaryNode<AnyType> t){if (t == null){return null;}if (t.right == null){return t;}return findMax(t.right);}/*** Internal method to remove from a subtree* @param x the item to remove* @param t the node that roots the subtree* @return the new root of the subtree*/private BinaryNode<AnyType> remove(AnyType x, BinaryNode<AnyType> t){if (t == null){return t; // Item not found ,do nothing}int compareResult = myCompare(x,t.element);if (compareResult < 0){t.left = remove(x,t.left);}else if (compareResult > 0){t.right = remove(x,t.right);}else if (t.left !=null && t.right!=null){//Two childrent.element = findMin(t.right).element;t.right = remove(t.element,t.right);}elset = (t.left !=null) ? t.left:t.right;return t;}/*** Internal method to insert into a subtree* @param x the item to insert* @param t the node that roots the subtree* @return the new root of the subtree*/private BinaryNode<AnyType> insert(AnyType x, BinaryNode<AnyType> t){if (t == null){return new BinaryNode<>(x,null,null);}int compareResult = myCompare(x,t.element);if (compareResult < 0){t.left = insert(x,t.left);}else if (compareResult > 0){t.right = insert(x,t.right);}else{//Duplicate; do nothing}return t;}}