10、二分搜索树

内容来自刘宇波老师算法与数据结构体系课

1、二分搜索树的结构

2、实现二分搜索树

深度优先
- 前序遍历：中左右
- 中序遍历：左中右，二分搜索树的中序遍历结果是顺序的
- 后序遍历：左右中，为二分搜索树释放内存
 
广度优先
- 层序遍历：更快找到问题的解，常用于算法设计中 - 无权图最短路径

/**
 * 二分搜索树: Binary Search Tree
 * 存储的元素必须可比较
 * 对重复元素不做处理
 * 增、删、查: 最差 O(n), 平均 O(h), h = logN
 */
public class BST<E extends Comparable<E>> {
 
    private class Node {
        public E e;
        public Node left;
        public Node right;
 
        public Node(E e) {
            this.e = e;
            this.left = null;
            this.right = null;
        }
    }
 
    private Node root;
    private int size;
 
    public BST() {
        root = null;
        size = 0;
    }
 
    public int size() {
        return size;
    }
 
    public boolean isEmpty() {
        return size == 0;
    }
 
    public void add(E e) {
        root = add(root, e);
    }
 
    /**
     * 向以 node 为根节点的二分搜索树中添加元素 e, 并返回新的根节点
     */
    private Node add(Node node, E e) {
        if (node == null) {
            size++;
            return new Node(e);
        }
 
        if (e.compareTo(node.e) < 0) node.left = add(node.left, e);
        else if (e.compareTo(node.e) > 0) node.right = add(node.right, e);
 
        return node;
    }
 
    public boolean contains(E e) {
        return contains(root, e);
    }
 
    /**
     * 在以 node 为根节点的二分搜索树中搜索是否存在元素 e
     */
    private boolean contains(Node node, E e) {
        if (node == null) return false;
 
        if (e.compareTo(node.e) == 0) return true;
        if (e.compareTo(node.e) < 0) return contains(node.left, e);
        return contains(node.right, e);
    }
 
    public E minimum() {
        if (size == 0) throw new RuntimeException("BST is empty!");
        return minimum(root).e;
    }
 
    /**
     * 返回以 node 为根节点的二分搜索树中的最小元素所在的节点
     */
    private Node minimum(Node node) {
        if (node.left == null) return node;
        return minimum(node.left);
    }
 
    public E maximum() {
        if (size == 0) throw new RuntimeException("BST is empty!");
        return maximum(root).e;
    }
 
    /**
     * 返回以 node 为根节点的二分搜索树中的最大元素所在的节点
     */
    private Node maximum(Node node) {
        if (node.right == null) return node;
        return maximum(node.right);
    }
 
    public E removeMin() {
        E min = minimum();
        root = removeMin(root);
        return min;
    }
 
    /**
     * 删除以 node 为根节点的二分搜索树的最小元素所在的节点, 并返回新的根节点
     */
    private Node removeMin(Node node) {
        if (node.left == null) {
            Node rightNode = node.right;
            node.right = null;
            size--;
            return rightNode;
        }
 
        node.left = removeMin(node.left);
        return node;
    }
 
    public E removeMax() {
        E max = maximum();
        root = removeMax(root);
        return max;
    }
 
    /**
     * 删除以 node 为根节点的二分搜索树的最大元素所在的节点, 并返回新的根节点
     */
    private Node removeMax(Node node) {
        if (node.right == null) {
            Node leftNode = node.left;
            node.left = null;
            size--;
            return leftNode;
        }
 
        node.right = removeMax(node.right);
        return node;
    }
 
    public void remove(E e) {
        root = remove(root, e);
    }
 
    /**
     * 以 node 为根节点的二分搜索树, 删除指定元素 e 所在的节点, 并返回新的根节点
     */
    private Node remove(Node node, E e) {
        if (node == null) return null;
 
        if (e.compareTo(node.e) < 0) {
            node.left = remove(node.left, e);
            return node;
        } else if (e.compareTo(node.e) > 0) {
            node.right = remove(node.right, e);
            return node;
        } else {
            if (node.left == null) {
                Node rightNode = node.right;
                node.right = null;
                size--;
                return rightNode;
            } else if (node.right == null) {
                Node leftNode = node.left;
                node.left = null;
                size--;
                return leftNode;
            } else {
                // 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, 它是后继节点
                // 用这个节点顶替待删除节点的位置
                Node successor = minimum(node.right);
                successor.right = removeMin(node.right); // 已经 size-- 了
                successor.left = node.left;
                node.left = node.right = null;
                return successor;
            }
        }
    }
 
    public void preOrder() {
        preOrder(root);
    }
 
    /**
     * 对以 node 为根节点的二分搜索树进行前序遍历
     */
    private void preOrder(Node node) {
        if (node == null) return;
 
        System.out.println(node.e);
        preOrder(node.left);
        preOrder(node.right);
    }
 
    public void inOrder() {
        inOrder(root);
    }
 
    /**
     * 对以 node 为根节点的二分搜索树进行中序遍历
     */
    private void inOrder(Node node) {
        if (node == null) return;
 
        inOrder(node.left);
        System.out.println(node.e);
        inOrder(node.right);
    }
 
    public void postOrder() {
        postOrder(root);
    }
 
    /**
     * 对以 node 为根节点的二分搜索树进行后序遍历
     */
    private void postOrder(Node node) {
        if (node == null) return;
 
        postOrder(node.left);
        postOrder(node.right);
        System.out.println(node.e);
    }
 
    /**
     * 二分搜索树的层序遍历
     */
    public void levelOrder() {
        Queue<Node> queue = new LoopQueue<>();
        queue.enqueue(root);
 
        Node cur;
        while (!queue.isEmpty()) {
            cur = queue.dequeue();
            System.out.println(cur.e);
            if (cur.left != null) queue.enqueue(cur.left);
            if (cur.right != null) queue.enqueue(cur.right);
        }
    }
 
    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        generateBSTString(root, 0, sb);
        return sb.toString();
    }
 
    /**
     * 生成以 node 为根节点, depth 为深度的描述二叉树的字符串
     */
    private void generateBSTString(Node node, int depth, StringBuilder sb) {
        if (node == null) {
            sb.append(generateDepthString(depth)).append("null\n");
            return;
        }
 
        sb.append(generateDepthString(depth)).append(node.e).append("\n");
        generateBSTString(node.left, depth + 1, sb);
        generateBSTString(node.right, depth + 1, sb);
    }
 
    private String generateDepthString(int depth) {
        return "--".repeat(Math.max(0, depth));
    }
 
    /**
     * 寻找 e 的 floor 值
     * 不存在时返回 null(e 比 BST 中的最小值还小)
     */
    public E floor(E e) {
        if (isEmpty() || e.compareTo(minimum()) < 0) return null;
        return floor(root, e).e;
    }
 
    /**
     * 在以 node 为根节点的二分搜索树中搜索元素 e 的 floor 节点
     */
    private Node floor(Node node, E e) {
        if (node == null) return null;
 
        if (node.e.compareTo(e) == 0) return node;
        if (node.e.compareTo(e) > 0) return floor(node.left, e);
 
        Node tempNode = floor(node.right, e);
        return tempNode != null ? tempNode : node;
    }
 
    /**
     * 寻找 e 的 ceil 值
     * 不存在时返回 null(e 比 BST 中的最大值还大)
     */
    public E ceil(E e) {
        if (isEmpty() || e.compareTo(maximum()) > 0) return null;
        return ceil(root, e).e;
    }
 
    /**
     * 在以 node 为根节点的二分搜索树中搜索元素 e 的 ceil 节点
     */
    private Node ceil(Node node, E e) {
        if (node == null) return null;
 
        if (node.e.compareTo(e) == 0) return node;
        if (node.e.compareTo(e) < 0) return ceil(node.right, e);
 
        Node tempNode = ceil(node.left, e);
        return tempNode != null ? tempNode : node;
    }
}

3、BST 集合

public interface Set<E> {
 
    void add(E e);
 
    void remove(E e);
 
    boolean contains(E e);
 
    int getSize();
 
    boolean isEmpty();
 
}

/**
 * 基于二分搜索树 BST 实现的 Set, 存储的元素必须可比较
 */
public class BSTSet<E extends Comparable<E>> implements Set<E> {
 
    private final BST<E> bst;
 
    public BSTSet() {
        bst = new BST<>();
    }
 
    @Override
    public void add(E e) {
        bst.add(e);
    }
 
    @Override
    public void remove(E e) {
        bst.remove(e);
    }
 
    @Override
    public boolean contains(E e) {
        return bst.contains(e);
    }
 
    @Override
    public int getSize() {
        return bst.size();
    }
 
    @Override
    public boolean isEmpty() {
        return bst.isEmpty();
    }
}

4、BST 映射

public interface Map<K, V> {
 
    void add(K key, V value);
 
    V remove(K key);
 
    boolean contains(K key);
 
    V get(K key);
 
    void set(K key, V newValue);
 
    int getSize();
 
    boolean isEmpty();
 
}

/**
 * key 不能重复, 且必须可比较
 */
public class BSTMap<K extends Comparable<K>, V> implements Map<K, V> {
 
    private class Node {
        public K key;
        public V value;
        public Node left;
        public Node right;
 
        public Node(K key, V value) {
            this.key = key;
            this.value = value;
            this.left = null;
            this.right = null;
        }
    }
 
    private Node root;
    private int size;
 
    public BSTMap() {
        root = null;
        size = 0;
    }
 
    /**
     * 返回以 node 为根节点的二分搜索树中, 键为 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 node;
        if (key.compareTo(node.key) < 0) return getNode(node.left, key);
        return getNode(node.right, key);
    }
 
    @Override
    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添加元素 (key, value), 并返回新的根节点
     */
    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;
 
        return node;
    }
 
    /**
     * 返回以 node 为根节点的二分搜索树中的最小元素所在的节点
     */
    private Node minimum(Node node) {
        if (node.left == null) return node;
        return minimum(node.left);
    }
 
    /**
     * 删除以 node 为根节点的二分搜索树的最小元素所在的节点, 并返回新的根节点
     */
    private Node removeMin(Node node) {
        if (node.left == null) {
            Node rightNode = node.right;
            node.right = null;
            size--;
            return rightNode;
        }
 
        node.left = removeMin(node.left);
        return node;
    }
 
    @Override
    public V remove(K key) {
        Node node = getNode(root, key);
        if (node != null) {
            root = remove(root, key);
            return node.value;
        }
        return null;
    }
 
    /**
     * 以 node 为根节点的二分搜索树, 删除键为 key 所在的节点, 并返回新的根节点
     */
    private Node remove(Node node, K key) {
        if (node == null) return null;
 
        if (key.compareTo(node.key) < 0) {
            node.left = remove(node.left, key);
            return node;
        } else if (key.compareTo(node.key) > 0) {
            node.right = remove(node.right, key);
            return node;
        } else {
            if (node.left == null) {
                Node rightNode = node.right;
                node.right = null;
                size--;
                return rightNode;
            } else if (node.right == null) {
                Node leftNode = node.left;
                node.left = null;
                size--;
                return leftNode;
            } else {
                Node successor = minimum(node.right);
                successor.right = removeMin(node.right);
                successor.left = node.left;
                node.left = node.right = null;
                return successor;
            }
        }
    }
 
    @Override
    public boolean contains(K key) {
        return getNode(root, key) != null;
    }
 
    @Override
    public V get(K key) {
        Node node = getNode(root, key);
        return node != null ? node.value : null;
    }
 
    @Override
    public void set(K key, V newValue) {
        Node node = getNode(root, key);
        if (node != null) node.value = newValue;
        else throw new IllegalArgumentException(key + " doesn't exist!");
    }
 
    @Override
    public int getSize() {
        return size;
    }
 
    @Override
    public boolean isEmpty() {
        return size == 0;
    }
}

5、链表集合

/**
 * 基于链表 LinkedList 实现的 Set
 */
public class LinkedListSet<E> implements Set<E> {
 
    private final LinkedList<E> list;
 
    public LinkedListSet() {
        list = new LinkedList<>();
    }
 
    @Override
    public void add(E e) {
        if (!list.contains(e)) list.addFirst(e);
    }
 
    @Override
    public void remove(E e) {
        list.removeElement(e);
    }
 
    @Override
    public boolean contains(E e) {
        return list.contains(e);
    }
 
    @Override
    public int getSize() {
        return list.getSize();
    }
 
    @Override
    public boolean isEmpty() {
        return list.isEmpty();
    }
}

6、链表映射

/**
 * key 不能重复
 */
public class LinkedListMap<K, V> implements Map<K, V> {
 
    private class Node {
        public K key;
        public V value;
        public Node next;
 
        public Node(K key, V value, Node next) {
            this.key = key;
            this.value = value;
            this.next = next;
        }
 
        public Node() {
            this(null, null, null);
        }
 
        @Override
        public String toString() {
            return key.toString() + " : " + value.toString();
        }
    }
 
    private final Node dummyHead;
    private int size;
 
    public LinkedListMap() {
        dummyHead = new Node();
        size = 0;
    }
 
    private Node getNode(K key) {
        Node cur = dummyHead.next;
        while (cur != null) {
            if (cur.key.equals(key)) return cur;
            cur = cur.next;
        }
        return null;
    }
 
    @Override
    public void add(K key, V value) {
        Node node = getNode(key);
        if (node == null) {
            dummyHead.next = new Node(key, value, dummyHead.next);
            size++;
        } else node.value = value;
    }
 
    @Override
    public V remove(K key) {
        Node prev = dummyHead;
        while (prev.next != null) {
            if (prev.next.key.equals(key)) {
                Node delNode = prev.next;
                prev.next = delNode.next;
                delNode.next = null;
                size--;
                return delNode.value;
            }
            prev = prev.next;
        }
 
        return null;
    }
 
    @Override
    public boolean contains(K key) {
        return getNode(key) != null;
    }
 
    @Override
    public V get(K key) {
        Node node = getNode(key);
        return node != null ? node.value : null;
    }
 
    @Override
    public void set(K key, V newValue) {
        Node node = getNode(key);
        if (node != null) node.value = newValue;
        else throw new IllegalArgumentException(key + " doesn't exist!");
    }
 
    @Override
    public int getSize() {
        return size;
    }
 
    @Override
    public boolean isEmpty() {
        return size == 0;
    }
}

7、二分搜索树非递归

栈和递归的密切关系

/**
 * Binary Search Tree Not Recursion
 */
public class BSTNR<E extends Comparable<E>> {
 
    private class Node {
        public E e;
        public Node left;
        public Node right;
 
        public Node(E e) {
            this.e = e;
            this.left = null;
            this.right = null;
        }
    }
 
    private Node root;
    private int size;
 
    public BSTNR() {
        root = null;
        size = 0;
    }
 
    public int size() {
        return size;
    }
 
    public boolean isEmpty() {
        return size == 0;
    }
 
    public void add(E e) {
        if (root == null) {
            root = new Node(e);
            size++;
            return;
        }
 
        Node parent = root;
        while (true) {
            if (e.compareTo(parent.e) == 0) return;
 
            if (e.compareTo(parent.e) < 0) {
                if (parent.left == null) {
                    parent.left = new Node(e);
                    size++;
                    return;
                }
                parent = parent.left;
            }
 
            if (e.compareTo(parent.e) > 0) {
                if (parent.right == null) {
                    parent.right = new Node(e);
                    size++;
                    return;
                }
                parent = parent.right;
            }
        }
    }
 
    public boolean contains(E e) {
        Node cur = root;
        while (cur != null) {
            if (e.compareTo(cur.e) == 0) return true;
            if (e.compareTo(cur.e) < 0) cur = cur.left;
            else cur = cur.right;
        }
        return false;
    }
 
    public void preOrder() {
        Stack<Node> stack = new ArrayStack<>();
        stack.push(root);
 
        Node cur;
        while (!stack.isEmpty()) {
            cur = stack.pop();
            System.out.println(cur.e);
            if (cur.right != null) stack.push(cur.right);
            if (cur.left != null) stack.push(cur.left);
        }
    }
 
    public void levelOrder() {
        Queue<Node> queue = new LoopQueue<>();
        queue.enqueue(root);
 
        Node cur;
        while (!queue.isEmpty()) {
            cur = queue.dequeue();
            System.out.println(cur.e);
            if (cur.left != null) queue.enqueue(cur.left);
            if (cur.right != null) queue.enqueue(cur.right);
        }
    }
}

8、更多话题

8.1、二分搜索树的顺序性

• minimum、maximun
• predecessor、successor
• floor、ceil
• rank、select：前序遍历可以实现

8.2、更多拓展

• 维护 size 的二分搜索树
• 维护 depth 的二分搜索树
• 支持重复元素的二分搜索树
• 维护 count 的二分搜索树