1.. 平衡二叉树
- 平衡二叉树要求，对于任意一个节点，左子树和右子树的高度差不能超过1。
- 平衡二叉树的高度和节点数量之间的关系也是O(logn)
- 为二叉树标注节点高度并计算平衡因子
- AVL树是一棵平衡二叉树
2.. 实现AVL树的业务逻辑
import java.util.ArrayList; public class AVLTree<K extends Comparable<K>, V> { private class Node {
public K key;
public V value;
public Node left;
public Node right;
public int height; // 构造函数
public Node(K key, V value) {
this.key = key;
this.value = value;
left = null;
right = null;
height = 1;
}
} private Node root;
private int size; // 构造函数
public AVLTree() {
root = null;
size = 0;
} // 实现getSize方法
public int getSize() {
return size;
} // 实现isEmpty方法
public boolean isEmpty() {
return size == 0;
} // 判断该二叉树是否为二分搜索树
public boolean isBST() {
ArrayList<K> keys = new ArrayList<>();
inOrder(root, keys);
for (int i = 1; i < keys.size(); i++) {
if (keys.get(i - 1).compareTo(keys.get(i)) > 0) {
return false;
}
}
return true;
} private void inOrder(Node node, ArrayList<K> keys) { if (node == null) {
return;
}
inOrder(node.left, keys);
keys.add(node.key);
inOrder(node.right, keys);
} // 判断二叉树是否为平衡二叉树
public boolean isBalanced() {
return isBalanced(root);
} // 判断以node为根的二叉树是否为平衡二叉树
private boolean isBalanced(Node node) { if (node == null) {
return true;
}
int balanceFactor = getBalanceFactor(node);
if (Math.abs(balanceFactor) > 1) {
return false;
}
return isBalanced(node.left) && isBalanced(node.right);
} // 返回节点node的高度值
private int getHeight(Node node) {
if (node == null) {
return 0;
}
return node.height;
} // 返回节点node的平衡因子
private int getBalanceFactor(Node node) {
if (node == null) {
return 0;
}
return getHeight(node.left) - getHeight(node.right);
} // 对节点y进行向右旋转操作，返回旋转后新的根节点x
// y x
// / \ / \
// x T4 向右旋转 (y) z y
// / \ - - - - - - - -> / \ / \
// z T3 T1 T2 T3 T4
// / \
// T1 T2
private Node rightRotate(Node y) { Node x = y.left;
Node T3 = x.right; // 向右旋转
x.right = y;
y.left = T3; // 更新height
y.height = Math.max(getHeight(y.left), getHeight(y.right)) + 1;
x.height = Math.max(getHeight(x.left), getHeight(x.right)) + 1; return x;
} // 对节点y进行向左旋转操作，返回旋转后新的根节点x
// y x
// / \ / \
// T1 x 向左旋转 (y) y z
// / \ - - - - - - - -> / \ / \
// T2 z T1 T2 T3 T4
// / \
// T3 T4
private Node leftRotate(Node y) { Node x = y.right;
Node T2 = x.left; // 向左旋转
x.left = y;
y.right = T2; //更新height
y.height = Math.max(getHeight(y.left), getHeight(y.right)) + 1;
x.height = Math.max(getHeight(x.left), getHeight(x.right)) + 1; return x;
} // 实现add方法
public void add(K key, V value) {
root = add(root, key, value);
} // 向以node为根节点的二分搜索树中插入元素(key, value)，递归算法
// 返回插入新元素后的二分搜索树的根
private Node add(Node node, K key, V value) { if (node == null) {
size++;
return new Node(key, value);
} if (key.compareTo(node.key) < 0) {
node.left = add(node.left, key, value);
} else if (key.compareTo(node.key) > 0) {
node.right = add(node.right, key, value);
} else {
node.value = value;
} // 更新height值
node.height = 1 + Math.max(getHeight(node.left), getHeight(node.right)); // 计算平衡因子
int balanceFactor = getBalanceFactor(node); // 平衡维护
// LL
if (balanceFactor > 1 && getBalanceFactor(node.left) >= 0) {
return rightRotate(node);
}
// RR
if (balanceFactor < -1 && getBalanceFactor(node.right) <= 0) {
return leftRotate(node);
} // LR
if (balanceFactor > 1 && getBalanceFactor(node.left) < 0) {
node.left = leftRotate(node.left);
return rightRotate(node);
}
// RL
if (balanceFactor < -1 && getBalanceFactor(node.right) > 0) {
node.right = rightRotate(node.right);
return leftRotate(node);
} return node;
} // 返回以node为根节点的二分搜索树中，key所在的节点
private Node getNode(Node node, K key) { if (node == null)
return null; if (key.compareTo(node.key) < 0) {
return getNode(node.left, key);
} else if (key.compareTo(node.key) > 0) {
return getNode(node.right, key);
} else {
return node;
}
} public boolean contains(K key) {
return getNode(root, key) != null;
} public V get(K key) { Node node = getNode(root, key);
return node == null ? null : node.value;
} public void set(K key, V newValue) {
Node node = getNode(root, key);
if (node == null)
throw new IllegalArgumentException(key + " doesn't exist!"); node.value = newValue;
} // 返回以node为根的二分搜索树的最小元素所在节点
private Node minimum(Node node) {
if (node.left == null) {
return node;
}
return minimum(node.left);
} // 实现remove方法
// 删除二分搜索树中键为key的节点
public V remove(K key) {
Node node = getNode(root, key); if (node != null) {
root = remove(root, key);
return node.value;
}
return null;
} // 删除以node为根节点的二分搜索树中键为key的节点，递归算法
// 返回删除节点后新的二分搜索树的根
private Node remove(Node node, K key) {
if (node == null) {
return null;
} Node retNode;
if (key.compareTo(node.key) < 0) {
node.left = remove(node.left, key);
retNode = node;
} else if (key.compareTo(node.key) > 0) {
node.right = remove(node.right, key);
retNode = node;
} else {
// 待删除节点左子树为空的情况
if (node.left == null) {
Node rightNode = node.right;
node.right = null;
size--;
retNode = rightNode;
// 待删除节点右子树为空的情况
} else if (node.right == null) {
Node leftNode = node.left;
node.left = null;
size--;
retNode = leftNode;
// 待删除节点左右子树均不为空
// 找到比待删除节点大的最小节点，即待删除节点右子树的最小节点
// 用这个节点顶替待删除节点
} else {
Node successor = minimum(node.right);
successor.right = remove(node.right, successor.key); //这里进行了size--操作
successor.left = node.left;
node.left = null;
node.right = null;
retNode = successor;
}
} if (retNode == null) {
return null;
} // 更新height值
retNode.height = 1 + Math.max(getHeight(retNode.left), getHeight(retNode.right)); // 计算平衡因子
int balanceFactor = getBalanceFactor(retNode); // 平衡维护
// LL
if (balanceFactor > 1 && getBalanceFactor(retNode.left) >= 0) {
return rightRotate(retNode);
}
// RR
if (balanceFactor < -1 && getBalanceFactor(retNode.right) <= 0) {
return leftRotate(retNode);
} // LR
if (balanceFactor > 1 && getBalanceFactor(retNode.left) < 0) {
node.left = leftRotate(retNode.left);
return rightRotate(retNode);
}
// RL
if (balanceFactor < -1 && getBalanceFactor(retNode.right) > 0) {
node.right = rightRotate(retNode.right);
return leftRotate(retNode);
} return retNode;
} // 打印测试
public static void main(String[] args) { System.out.println("Pride and Prejudice"); ArrayList<String> words = new ArrayList<>(); if (FileOperation.readFile("pride-and-prejudice.txt", words)) { System.out.println("Total words: " + words.size()); AVLTree<String, Integer> map = new AVLTree<>();
for (String word : words) {
if (map.contains(word)) {
map.set(word, map.get(word) + 1);
} else {
map.add(word, 1);
}
} System.out.println("Total different words: " + map.getSize());
System.out.println("Frequency of PRIDE: " + map.get("pride"));
System.out.println("Frequency of PREJUDICE: " + map.get("prejudice")); System.out.println("is BST: " + map.isBST()); System.out.println("is Balanced: " + map.isBalanced());
}
}
}