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目录
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\(\mathcal{O}(\sqrt{N})\),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\(\mathcal{O}(log N)\)。
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\(\mathcal{O}(log N)\)，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
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;
}
}