二叉搜索树(Binary Search Tree)（Java实现）

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二叉搜索树(Binary Search Tree)（Java实现）

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 $O (\sqrt{N})$ ,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 $O (l o g N)$ 。

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 $O (l o g N)$ ，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 node
        BinaryNode<AnyType> left; // Left child
        BinaryNode<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

在这里插入图片描述

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 children
            t.element = findMin(t.right).element;
            t.right = remove(t.element,t.right);
        }
        else
            t = (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 node
        BinaryNode<AnyType> left; // Left child
        BinaryNode<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 children
            t.element = findMin(t.right).element;
            t.right = remove(t.element,t.right);
        }
        else
            t = (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;
    }

}