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AVL树又被叫做平衡二叉搜索树、平衡二叉树。AVL是其发明者的首字母缩写。

这篇文章中，AVLTreeMap 类集成了 java.util.Map 接口，并利用 AVL 树结构实现了 Map 接口的所有方法。本文还给出了测试代码。

为什么要发明AVL树？

当我按照从小到大或者从大到小的顺序向二叉查找树插入节点，二叉查找树就会退化成一个链表。这是二叉查找树的最差情况。搜索、插入、删除的最差效率都是 O(N)。这样就失去了用二叉查找树优化查找方法的意义。

就算不是最坏情况，也会出现非常不平衡的树，造成查找效率大于 O(logN) 小于 O(N) 。注意这里 logN 是以2为底N的对数。

AVL树的时间复杂度

因为含有数学公式，就使用图片了。

代码实现

AVLTreeMap.java

package zhangchao.avl;import java.util.*;/*** 利用AVL树，也就是平衡二叉树来实现map* @author zhangchao* @param <K> 键* @param <V> 值*/
public class AVLTreeMap<K,V> implements Map<K, V>{// 根节点private Node<K, V> root = null;private Comparator<K> comparator;public AVLTreeMap(Comparator<K> comparator) {this.comparator = comparator;}@Overridepublic int size() {if (null == root) {return 0;}int size = 0;// 当前层的节点列表List<Node<K, V>> currentLevel = new ArrayList<>();currentLevel.add(root);while (!currentLevel.isEmpty()) {size += currentLevel.size();// 下一层的节点列表List<Node<K, V>> nextLevel = new ArrayList<>();for (Node<K, V> tmpNode : currentLevel) {if (null != tmpNode.leftChild) {nextLevel.add(tmpNode.leftChild);}if (null != tmpNode.rightChild) {nextLevel.add(tmpNode.rightChild);}}currentLevel.clear();currentLevel.addAll(nextLevel);}return size;}@Overridepublic boolean isEmpty() {return (null == root);}@Overridepublic boolean containsKey(Object keyObj) {if (null == root) {return false;}K key = (K)keyObj;Node<K, V> current = this.root;while(null != current) {int compareResult = this.comparator.compare(key, current.key);if (compareResult < 0) {current = current.leftChild;} else if (compareResult > 0) {current = current.rightChild;} else {return true;}}return false;}@Overridepublic boolean containsValue(Object value) {if (null == this.root) {return false;}List<Node<K, V>> nodeList = this.nodeList();for (Node<K, V> node : nodeList) {if (null == value && null == node.value) {return true;}if (null != value && value.equals(node.value)) {return true;}}return false;}@Overridepublic V get(Object keyObj) {if (null == this.root) {return null;}K key = (K)keyObj;Node<K, V> current = this.root;while(null != current) {int compareResult = this.comparator.compare(key, current.key);if (compareResult < 0) {current = current.leftChild;} else if (compareResult > 0) {current = current.rightChild;} else {return current.value;}}return null;}/*** 右旋操作* @param parent 爷爷节点* @param y 父级节点* @param x 子级节点*/private void rotateRight(Node<K, V> parent, Node<K, V> y, Node<K, V> x) {y.leftChild = x.rightChild;x.rightChild = y;if (null == parent) {this.root = x;} else {// 判断原来的y是parent的左子节点还是右子节点。if (null != parent.leftChild && 0 == this.comparator.compare(parent.leftChild.key, y.key)) {parent.leftChild = x;} else if (null != parent.rightChild && 0 == this.comparator.compare(parent.rightChild.key, y.key)) {parent.rightChild = x;}}y.height = this.calHeight(y);x.height = this.calHeight(x);if (null != parent) {parent.height = this.calHeight(parent);}}/*** 左旋操作* @param parent 爷爷节点* @param y 父级节点* @param x 子级节点*/private void rotateLeft(Node<K, V> parent, Node<K, V> y, Node<K, V> x) {y.rightChild = x.leftChild;x.leftChild = y;if (null == parent) {this.root = x;} else {// 判断原来的y是parent的左子节点还是右子节点。if (null != parent.leftChild && 0 == this.comparator.compare(parent.leftChild.key, y.key)) {parent.leftChild = x;} else if (null != parent.rightChild && 0 == this.comparator.compare(parent.rightChild.key, y.key)) {parent.rightChild = x;}}y.height = this.calHeight(y);x.height = this.calHeight(x);if (null != parent) {parent.height = this.calHeight(parent);}}@Overridepublic V put(K key, V value) {if (null == this.root) {this.root = new Node<>();this.root.key = key;this.root.value = value;this.root.height = 1;return null;}// 如果key是全新的，保存所有的父亲节点。List<Node<K, V>> linkList = new ArrayList<>();// 如果key是全新的，这个变量是新节点的父亲节点。Node<K, V> parent = null;Node<K, V> current = root;int compareResult = 0;while (null != current) {compareResult = this.comparator.compare(key, current.key);if (compareResult < 0) {parent = current;linkList.add(parent);current = current.leftChild;} else if (compareResult > 0) {parent = current;linkList.add(parent);current = current.rightChild;} else {// 有相等的key，直接设置值就可以了。V oldValue = current.value;current.value = value;return oldValue;}}Node<K, V> newItem = new Node<K, V>();newItem.key = key;newItem.value = value;newItem.height = 1;if (compareResult < 0) {parent.leftChild = newItem;} else if (compareResult > 0) {parent.rightChild = newItem;}// 更新祖先节点的高度final int size = linkList.size();for (int i = size - 1; i >= 0; i--) {Node<K, V> item = linkList.get(i);item.height = calHeight(item);}linkList.add(newItem);int parentSize = linkList.size();for (int i = parentSize - 1; i >= 0; i--) {// 当前节点Node<K, V> z = linkList.get(i);// z的父节点，如果z是根节点，那么就是null。Node<K, V> z_parent = null;if (i > 0) {z_parent = linkList.get(i - 1);}int leftHeight = this.calHeight(z.leftChild);int rightHeight = this.calHeight(z.rightChild);int balance = leftHeight - rightHeight;if (balance > 1) { // LL 或 LRNode<K, V> y = z.leftChild;Node<K, V> x = linkList.get(i + 2);boolean isLL = (null != y.leftChild  && this.comparator.compare(y.leftChild.key, x.key) == 0);boolean isLR = (null != y.rightChild && this.comparator.compare(y.rightChild.key, x.key) == 0);if (isLL) { // LL 右旋this.rotateRight(z_parent, z, y);}else if (isLR) {    // LR// y和x之间左旋this.rotateLeft(z, y, x);// z和x之间右旋this.rotateRight(z_parent, z, x);}break; // 停止for循环} else if (balance < -1) { // RR 或 RLNode<K, V> y = z.rightChild;Node<K, V> x = linkList.get(i + 2);boolean isRR = (null != y.rightChild && this.comparator.compare(y.rightChild.key, x.key) == 0);boolean isRL = (null != y.leftChild && this.comparator.compare(y.leftChild.key, x.key) == 0);if (isRR) {this.rotateLeft(z_parent, z, y);} else if (isRL) {// y和x之间右旋this.rotateRight(z, y, x);// z和x之间左旋this.rotateLeft(z_parent, z, x);}break; // 停止for循环}}// 更新祖先节点高度for (int i = parentSize - 1; i >= 0; i--) {Node<K, V> item = linkList.get(i);item.height = calHeight(item);}return null;}private List<Node<K,V>> getNodeAndParent(K key, List<Node<K, V>> parents) {if (null == this.root) {return null;}Node<K, V> parent = null;Node<K, V> current = this.root;while(null != current) {int compareResult = this.comparator.compare(key, current.key);if (compareResult < 0) {parent = current;if (null != parents) {parents.add(parent);}current = current.leftChild;} else if (compareResult > 0) {parent = current;if (null != parents) {parents.add(parent);}current = current.rightChild;} else {List<Node<K, V>> result = new ArrayList<>();result.add(current);result.add(parent);return result;}}return null;}private K deleteAsBST(Node<K, V> node, Node<K, V> parent) {K endKey = null;// 叶子节点if (null == node.leftChild && null == node.rightChild) {if (node == parent.leftChild) {parent.leftChild = null;} else {parent.rightChild = null;}return parent.key;}// 左子节点为空，只有右子节点else if (null == node.leftChild && null != node.rightChild) {if (node == this.root) {this.root = node.rightChild;} else if (node == parent.leftChild) {parent.leftChild = node.rightChild;} else if (node == parent.rightChild) {parent.rightChild = node.rightChild;}endKey = node.rightChild.key;node.rightChild = null;return endKey;}// else 包含两种情况：// 1.左子节点不为空，右子为空// 2.左子节点不为空，右子不为空// 要删除的节点的左子树中，找出最大节点。Node<K, V> current = node.leftChild;Node<K, V> currentParent = node;while (null != current.rightChild) {currentParent = current;current = current.rightChild;}// 把current从原位置删除if (current == currentParent.leftChild) {currentParent.leftChild = current.leftChild;} else if (current == currentParent.rightChild) {currentParent.rightChild = current.leftChild;}// 让current取代node的位置if (node == this.root) {this.root = current;} else if (node == parent.leftChild) {parent.leftChild = current;} else {parent.rightChild = current;}current.leftChild = node.leftChild;current.rightChild = node.rightChild;node.leftChild = null;node.rightChild = null;if (null == current.leftChild) {return current.key;} else {Node<K, V> p1 = current.leftChild;while (null != p1.rightChild) {p1 = p1.rightChild;}return p1.key;}}@Overridepublic V remove(Object keyObj) {// 空map，不执行删除操作。if (null == this.root) {return null;}K key = (K)keyObj;// 只有根节点的情况if (null == this.root.leftChild && null == this.root.rightChild) {if (this.comparator.compare(key ,this.root.key) == 0) {V v = this.root.value;this.root = null;return v;} else {return null;}}// 不包含key就返回nullList<Node<K, V>> nodeAndParent = this.getNodeAndParent(key, new ArrayList<>());// map中没有对应的key，不执行删除操作。if (null == nodeAndParent || nodeAndParent.isEmpty()) {return null;}Node<K, V> node = nodeAndParent.get(0); // 要删除的节点V result = node.value;Node<K, V> parent = nodeAndParent.get(1); // 要删除的节点的父亲节点// 按照二叉搜索树（BST）的方式删除节点。返回结束节点的键。K endKey = this.deleteAsBST(node, parent);// 包含所有可能改动过高度的节点的列表。// 顺序是从根节点向下。// 替换了已删除节点位置的节点称为替换节点。// pathList的内容有以下三种情况：// 1. 叶子节点，pathList包含根节点到父节点。// 2. 没有左子节点，只有右子节点，pathList包含根节点到替换节点。// 3. 有左子节点，pathList包含根节点到替换节点，再加上替换节点到替换节点左子树最大节点。List<Node<K, V>> pathList = new ArrayList<>();List<Node<K,V>> endKeyResult = this.getNodeAndParent(endKey, pathList);pathList.add(endKeyResult.get(0));// 因为可能加入了节点，所以要重新计算 parents 的长度int size = pathList.size();for (int i = size - 1; i >= 0; i--) {Node<K, V> z_parent = i > 0 ? pathList.get(i - 1) : null;Node<K, V> z = pathList.get(i);// 更新高度z.height = this.calHeight(z);if (null != z_parent) {z_parent.height = this.calHeight(z_parent);}int leftHeight = calHeight(z.leftChild);int rightHeight = calHeight(z.rightChild);int balance = leftHeight - rightHeight;if (balance > 1) {Node<K, V> y = z.leftChild;Node<K, V> x = null;int y_leftHeight = calHeight(y.leftChild);int y_rightHeight = calHeight(y.rightChild);if (y_leftHeight >= y_rightHeight) {// LLx = y.leftChild;// z和y之间右旋this.rotateRight(z_parent, z, y);} else {// LRx = y.rightChild;// y和x之间左旋this.rotateLeft(z, y, x);// z和x之间右旋this.rotateRight(z_parent, z, x);}} else if (balance < -1) {Node<K, V> y = z.rightChild;Node<K, V> x = null;int y_leftHeight = calHeight(y.leftChild);int y_rightHeight = calHeight(y.rightChild);if (y_leftHeight >= y_rightHeight) {// RLx = y.leftChild;// y和x之间右旋this.rotateRight(z, y, x);// z和x之间左旋this.rotateLeft(z_parent, z, x);} else {// RRx = y.rightChild;// z和y之间左旋this.rotateLeft(z_parent, z, y);}}}return result;}// end public V remove(Object keyObj)@Overridepublic void putAll(Map<? extends K, ? extends V> m) {if (null == m) {return;}Set<? extends K> keySet = m.keySet();for (K key : keySet) {this.put(key, m.get(key));}}@Overridepublic void clear() {this.root = null;}private List<Node<K, V>> nodeList() {if (null == this.root) {return new ArrayList<Node<K, V>>();}List<Node<K, V>> result = new ArrayList<>();Stack<Node<K, V>> stack = new Stack<>();Node<K, V> current = this.root;while(null != current || !stack.isEmpty()) {while (null != current) {stack.push(current);current = current.leftChild;}current = stack.pop();// 放入结果列表中result.add(current);current = current.rightChild;}return result;}@Overridepublic Set<K> keySet() {List<Node<K, V>> nodeList = nodeList();Set<K> set = new TreeSet<>(this.comparator);for (Node<K, V> node : nodeList) {set.add(node.key);}return set;}@Overridepublic Collection<V> values() {List<Node<K, V>> nodeList = nodeList();List<V> result = new ArrayList<>();for (Node<K,V> node : nodeList) {result.add(node.value);}return result;}@Overridepublic Set<Entry<K, V>> entrySet() {List<Node<K, V>> nodeList = this.nodeList();Set<Entry<K, V>> set = new TreeSet<Entry<K, V>>((o1, o2) -> {Node<K, V> n1 = (Node<K, V>) o1;Node<K, V> n2 = (Node<K, V>) o2;return comparator.compare(n1.key, n2.key);});for (Node<K,V> node : nodeList) {set.add(node);}return set;}private int calHeightForCheck(Node<K,V> node) {if (null == node) {return 0;}int height = 0;List<Node<K,V>> currentLevel = new ArrayList<>();currentLevel.add(node);while (!currentLevel.isEmpty()) {height ++;List<Node<K,V>> nextLevel = new ArrayList<>();for (Node<K,V> tmpNode : currentLevel) {if (null != tmpNode.leftChild) {nextLevel.add(tmpNode.leftChild);}if (null != tmpNode.rightChild) {nextLevel.add(tmpNode.rightChild);}}currentLevel = nextLevel;}return height;}private void showTree(Node node, Node parent, int level, String prefix) {if (null == node) {return;}StringBuilder sb = new StringBuilder();for (int i = 0; i < level; i++) {sb.append("  ");}sb.append(prefix);sb.append(node.key).append("         ");if (parent != null) {sb.append(parent.key);}int balance = calHeightForCheck(node.leftChild) - calHeightForCheck(node.rightChild);sb.append("    ").append(balance);System.out.println(sb);level++;showTree(node.leftChild, node, level, "left : ");showTree(node.rightChild, node, level, "right: ");}/*** 打印树形结构。*/public void showTree() {if (null == root) {System.out.println("null");}showTree(root, null, 0, "root: ");}private void checkTree(Node node, Node parent, int level) {if (null == node) {return;}int balance = calHeightForCheck(node.leftChild) - calHeightForCheck(node.rightChild);if (balance < -1 || balance > 1) {throw new RuntimeException("balance < -1 || balance > 1");}level++;checkTree(node.leftChild, node, level);checkTree(node.rightChild, node, level);}/*** 检查树是不是符合AVL树的要求*/public void checkTree() {if (null == root) {return;}checkTree(root, null, 0);}/*** 以node为根节点，计算树的高度* @param node 根节点* @return 树的高度*/private int calHeight(Node<K,V> node) {if (null == node) {return 0;}int leftHeight = (null == node.leftChild) ? 0 : node.leftChild.height;int rightHeight = (null == node.rightChild) ? 0 : node.rightChild.height;return Math.max(leftHeight, rightHeight) + 1;}class Node<K,V> implements Entry<K,V> {K key = null;V value = null;int height;Node<K, V> leftChild;Node<K, V> rightChild;@Overridepublic K getKey() {return key;}@Overridepublic V getValue() {return value;}@Overridepublic V setValue(V tmpValue) {V oldValue = value;value = tmpValue;return oldValue;}public int getHeight() {return height;}}
}

测试代码 TestAVLTreeMap.java
这里面有和二叉查找树Map的对比。
二叉查找树Map的实现文章：https://blog.csdn.net/zhangchao19890805/article/details/128609922?spm=1001.2014.3001.5502

package zhangchao.avl.test;import zhangchao.avl.AVLTreeMap;import zhangchao.bst.BstTreeMap;import java.util.*;public class TestAVLTreeMap {public static void main(String[] args) {t7();}public static void t7() {int a[] = {20, 10, 21, 22, 5, 15, 1};Comparator<Integer> comparator = (o1, o2) ->{if (null == o1 && null == o2) {return 0;}if (null == o1 && null != o2) {return -1;}if (null != o1 && null == o2) {return 1;}return o1 - o2;};AVLTreeMap<Integer, String> avlTreeMap = new AVLTreeMap<>(comparator );for (int key : a) {avlTreeMap.put(key, "__" + key);}avlTreeMap.showTree();avlTreeMap.remove(20);System.out.println("\n");avlTreeMap.showTree();avlTreeMap.checkTree();}public static void t6() {Comparator<Integer> comparator = (o1, o2) ->{if (null == o1 && null == o2) {return 0;}if (null == o1 && null != o2) {return -1;}if (null != o1 && null == o2) {return 1;}return o1 - o2;};AVLTreeMap<Integer, String> avlTreeMap = new AVLTreeMap<>(comparator );BstTreeMap<Integer, String> bstTreeMap = new BstTreeMap<>(comparator);long t1;long t2;// 比对插入System.out.println("insert");Random r = new Random();final int MAX = 100000;List<Integer> list = new ArrayList<>();for (int i = 0; i < MAX; i++) {int key = r.nextInt(MAX);list.add(i);}t1 = System.currentTimeMillis();for (int key : list) {avlTreeMap.put(key, "__" + key);}t2 = System.currentTimeMillis();System.out.println("AVL:" + (t2 - t1));t1 = System.currentTimeMillis();for (int key : list) {bstTreeMap.put(key, "__" + key);}t2 = System.currentTimeMillis();System.out.println("BST:" + (t2 - t1));// 比对查询System.out.println("\nsearch");t1 = System.currentTimeMillis();for (int i = 0; i < MAX; i++) {avlTreeMap.get(i);}t2 = System.currentTimeMillis();System.out.println("AVL:" + (t2 - t1));t1 = System.currentTimeMillis();for (int i = 0; i < MAX; i++) {bstTreeMap.get(i);}t2 = System.currentTimeMillis();System.out.println("BST:" + (t2 - t1));
//        avlTreeMap.showTree();//      比对删除System.out.println("\nremove");t1 = System.currentTimeMillis();Collections.shuffle(list);for (int key : list) {avlTreeMap.remove(key);}t2 = System.currentTimeMillis();System.out.println("AVL:" + (t2 - t1));t1 = System.currentTimeMillis();for (int key : list) {bstTreeMap.remove(key);}t2 = System.currentTimeMillis();System.out.println("BST:" + (t2 - t1));avlTreeMap.checkTree();}public static void t3() {Map<Integer, String> map = new AVLTreeMap<>( (o1, o2) ->{if (null == o1 && null == o2) {return 0;}if (null == o1 && null != o2) {return -1;}if (null != o1 && null == o2) {return 1;}return o1 - o2;});int[] arr = new int[]{20,10,21,5,15,22,13,16};for (int i : arr) {map.put(i, "__" + String.valueOf(i));}AVLTreeMap avlTreeMap = (AVLTreeMap) map;avlTreeMap.showTree();avlTreeMap.remove(10);System.out.println("\n");avlTreeMap.showTree();avlTreeMap.checkTree();}public static void t2() {Map<Integer, String> map = new AVLTreeMap<>( (o1, o2) ->{if (null == o1 && null == o2) {return 0;}if (null == o1 && null != o2) {return -1;}if (null != o1 && null == o2) {return 1;}return o1 - o2;});int[] arr = new int[]{8,3,6,1,2,98,2,6,150,170,160,7,52,28,75,14,40,86,10,21,46,25};for (int i : arr) {map.put(i, "__" + String.valueOf(i));}AVLTreeMap avlTreeMap = (AVLTreeMap) map;avlTreeMap.showTree();avlTreeMap.remove(7);System.out.println("\n\n\n");avlTreeMap.showTree();avlTreeMap.checkTree();}public static void t1() {Map<Integer, String> map = new AVLTreeMap<>( (o1, o2) ->{if (null == o1 && null == o2) {return 0;}if (null == o1 && null != o2) {return -1;}if (null != o1 && null == o2) {return 1;}return o1 - o2;});int[] arr = new int[]{8,3,6,1,2,98,2,6,150,170,160,7,52,28,75,14,40,86,10,21,46,25};for (int i : arr) {map.put(i, "__" + String.valueOf(i));}AVLTreeMap avlTreeMap = (AVLTreeMap) map;avlTreeMap.showTree();System.out.println(map.get(3));System.out.println(map.get(6));System.out.println(map.get(98));System.out.println(map.get(null));Set<Integer> set = avlTreeMap.keySet();for (Integer i : set) {System.out.println(i);}System.out.println();HashSet<Integer> hashSet = new HashSet<>();for (int i : arr) {hashSet.add(i);}for (int i : hashSet) {if (!set.contains(i)) {System.out.println(false);}}System.out.println(set.size() + "   " + hashSet.size());System.out.println("containsKey 3: " + avlTreeMap.containsKey(3));System.out.println("containsKey 4: " + avlTreeMap.containsKey(4));System.out.println("containsValue __3: " + avlTreeMap.containsValue("__3"));System.out.println("containsValue __4: " + avlTreeMap.containsValue("__4"));System.out.println();Set<Map.Entry<Integer, String>> entrySet = avlTreeMap.entrySet();for (Map.Entry<Integer, String> item : entrySet) {System.out.println(item.getKey() + ": " + item.getValue());}avlTreeMap.checkTree();}
}