平衡二叉树的定义

AVL树

  • 继承结构

添加导致的失衡

四种旋转调整失衡

LL – 右旋转（单旋）

RR – 左旋转（单旋）

LR – RR左旋转，LL右旋转（双旋）

RL – LL右旋转，RR左旋转（双旋）

添加之后的修复

protected void afterAdd(Node<E> node) {
    while ((node = node.parent) != null) {
        if (isBalanced(node)) {
            // 更新高度
            updateHeight(node);
        } else {
            // 恢复平衡
            rebalance(node);
        }
    }
}


/**
 * 恢复平衡
 *
 * @param grand 第一个不平衡的父节点
 */
private void rebalance(Node<E> grand) {
    Node<E> parent = ((AVLNode<E>) grand).tallerChild();
    Node<E> node = ((AVLNode<E>) parent).tallerChild();
    if (parent.isLeftChild()) { // L
        if (node.isLeftChild()) { // LL
            rotateRight(grand);
        } else { // LR
            rotateLeft(parent);
            rotateRight(grand);
        }
    } else { // R
        if (node.isLeftChild()) { // RL
            rotateRight(parent);
            rotateLeft(grand);
        } else { // RR
            rotateLeft(grand);
        }
    }
}

旋转

private void rotateLeft(Node<E> grand) {
    Node<E> parent = grand.right;
    Node<E> child = parent.left;
    grand.right = child;
    parent.left = grand;
    afterRotate(grand, parent, child);
}

private void rotateRight(Node<E> grand) {
    Node<E> parent = grand.left;
    Node<E> child = parent.right;
    grand.left = child;
    parent.right = grand;
    afterRotate(grand, parent, child);
}

private void afterRotate(Node<E> grand, Node<E> parent, Node<E> child) {
    // 让parent称为子树的根节点
    parent.parent = grand.parent;
    if (grand.isLeftChild()) {
        grand.parent.left = parent;
    } else if (grand.isRightChild()) {
        grand.parent.right = parent;
    } else { // grand是root节点
        root = parent;
    }

    // 更新child的parent
    if (child != null) {
        child.parent = grand;
    }

    // 更新grand的parent
    grand.parent = parent;

    // 更新高度
    updateHeight(grand);
    updateHeight(parent);
}

例子

统一所有旋转操作

四种旋转操作最后的结果都是一样的，按照a,b,c,d,e,f,g大小来排列

private void rebalance(Node<E> grand) {
    Node<E> parent = ((AVLNode<E>) grand).tallerChild();
    Node<E> node = ((AVLNode<E>) parent).tallerChild();
    if (parent.isLeftChild()) { // L
        if (node.isLeftChild()) { // LL
            rotate(grand, node, node.right, parent, parent.right, grand);
        } else { // LR
            rotate(grand, parent, node.left, node, node.right, grand);
        }
    } else { // R
        if (node.isLeftChild()) { // RL
            rotate(grand, grand, node.left, node, node.right, parent);
        } else { // RR
            rotate(grand, grand, parent.left, parent, node.left, node);
        }
    }
}

private void rotate(
        Node<E> r, // 子树的根节点
        Node<E> b, Node<E> c,
        Node<E> d,
        Node<E> e, Node<E> f) {
    // 让d成为这棵子树的根节点
    d.parent = r.parent;
    if (r.isLeftChild()) {
        r.parent.left = d;
    } else if (r.isRightChild()) {
        r.parent.right = d;
    } else {
        root = d;
    }

    //b-c
    b.right = c;
    if (c != null) {
        c.parent = b;
    }
    updateHeight(b);

    // e-f
    f.left = e;
    if (e != null) {
        e.parent = f;
    }
    updateHeight(f);

    // b-d-f
    d.left = b;
    d.right = f;
    b.parent = d;
    f.parent = d;
    updateHeight(d);
}

删除导致的失衡

删除之后的修复

protected void afterRemove(Node<E> node) {
    while ((node = node.parent) != null) {
        if (isBalanced(node)) {
            // 更新高度
            updateHeight(node);
        } else {
            // 恢复平衡
            rebalance(node);
        }
    }
}

总结

总代码

  • AVLTree

    import java.util.Comparator;

public class AVLTree<E> extends BST<E> {
    public AVLTree() {
        this(null);
    }

    public AVLTree(Comparator<E> comparator) {
        super(comparator);
    }

    @Override
    protected void afterAdd(Node<E> node) {
        while ((node = node.parent) != null) {
            if (isBalanced(node)) {
                // 更新高度
                updateHeight(node);
            } else {
                // 恢复平衡
                rebalance(node);
                // 整棵树恢复平衡
                break;
            }
        }
    }

    @Override
    protected void afterRemove(Node<E> node) {
        while ((node = node.parent) != null) {
            if (isBalanced(node)) {
                // 更新高度
                updateHeight(node);
            } else {
                // 恢复平衡
                rebalance(node);
            }
        }
    }

    @Override
    protected Node<E> createNode(E element, Node<E> parent) {
        return new AVLNode<>(element, parent);
    }

    /**
     * 恢复平衡
     *
     * @param grand 高度最低的那个不平衡节点
     */
    @SuppressWarnings("unused")
    private void rebalance2(Node<E> grand) {
        Node<E> parent = ((AVLNode<E>) grand).tallerChild();
        Node<E> node = ((AVLNode<E>) parent).tallerChild();
        if (parent.isLeftChild()) { // L
            if (node.isLeftChild()) { // LL
                rotateRight(grand);
            } else { // LR
                rotateLeft(parent);
                rotateRight(grand);
            }
        } else { // R
            if (node.isLeftChild()) { // RL
                rotateRight(parent);
                rotateLeft(grand);
            } else { // RR
                rotateLeft(grand);
            }
        }
    }

    /**
     * 恢复平衡
     *
     * @param grand 高度最低的那个不平衡节点
     */
    private void rebalance(Node<E> grand) {
        Node<E> parent = ((AVLNode<E>) grand).tallerChild();
        Node<E> node = ((AVLNode<E>) parent).tallerChild();
        if (parent.isLeftChild()) { // L
            if (node.isLeftChild()) { // LL
                rotate(grand, node, node.right, parent, parent.right, grand);
            } else { // LR
                rotate(grand, parent, node.left, node, node.right, grand);
            }
        } else { // R
            if (node.isLeftChild()) { // RL
                rotate(grand, grand, node.left, node, node.right, parent);
            } else { // RR
                rotate(grand, grand, parent.left, parent, node.left, node);
            }
        }
    }

    private void rotate(
            Node<E> r, // 子树的根节点
            Node<E> b, Node<E> c,
            Node<E> d,
            Node<E> e, Node<E> f) {
        // 让d成为这棵子树的根节点
        d.parent = r.parent;
        if (r.isLeftChild()) {
            r.parent.left = d;
        } else if (r.isRightChild()) {
            r.parent.right = d;
        } else {
            root = d;
        }

        //b-c
        b.right = c;
        if (c != null) {
            c.parent = b;
        }
        updateHeight(b);

        // e-f
        f.left = e;
        if (e != null) {
            e.parent = f;
        }
        updateHeight(f);

        // b-d-f
        d.left = b;
        d.right = f;
        b.parent = d;
        f.parent = d;
        updateHeight(d);
    }

    private void rotateLeft(Node<E> grand) {
        Node<E> parent = grand.right;
        Node<E> child = parent.left;
        grand.right = child;
        parent.left = grand;
        afterRotate(grand, parent, child);
    }

    private void rotateRight(Node<E> grand) {
        Node<E> parent = grand.left;
        Node<E> child = parent.right;
        grand.left = child;
        parent.right = grand;
        afterRotate(grand, parent, child);
    }

    private void afterRotate(Node<E> grand, Node<E> parent, Node<E> child) {
        // 让parent称为子树的根节点
        parent.parent = grand.parent;
        if (grand.isLeftChild()) {
            grand.parent.left = parent;
        } else if (grand.isRightChild()) {
            grand.parent.right = parent;
        } else { // grand是root节点
            root = parent;
        }

        // 更新child的parent
        if (child != null) {
            child.parent = grand;
        }

        // 更新grand的parent
        grand.parent = parent;

        // 更新高度
        updateHeight(grand);
        updateHeight(parent);
    }

    private boolean isBalanced(Node<E> node) {
        return Math.abs(((AVLNode<E>) node).balanceFactor()) <= 1;
    }

    private void updateHeight(Node<E> node) {
        ((AVLNode<E>) node).updateHeight();
    }

    private static class AVLNode<E> extends Node<E> {
        int height = 1;

        public AVLNode(E element, Node<E> parent) {
            super(element, parent);
        }

        public int balanceFactor() {
            int leftHeight = left == null ? 0 : ((AVLNode<E>) left).height;
            int rightHeight = right == null ? 0 : ((AVLNode<E>) right).height;
            return leftHeight - rightHeight;
        }

        public void updateHeight() {
            int leftHeight = left == null ? 0 : ((AVLNode<E>) left).height;
            int rightHeight = right == null ? 0 : ((AVLNode<E>) right).height;
            height = 1 + Math.max(leftHeight, rightHeight);
        }

        public Node<E> tallerChild() {
            int leftHeight = left == null ? 0 : ((AVLNode<E>) left).height;
            int rightHeight = right == null ? 0 : ((AVLNode<E>) right).height;
            if (leftHeight > rightHeight) return left;
            if (leftHeight < rightHeight) return right;
            return isLeftChild() ? left : right;
        }

        @Override
        public String toString() {
            String parentString = "null";
            if (parent != null) {
                parentString = parent.element.toString();
            }
            return element + "_p(" + parentString + ")_h(" + height + ")";
        }
    }
}

  • BST

    @SuppressWarnings("unchecked")
public class BST<E> extends BinaryTree<E> {
    private Comparator<E> comparator;

    public BST() {
        this(null);
    }

    public BST(Comparator<E> comparator) {
        this.comparator = comparator;
    }

    public void add(E element) {
        elementNotNullCheck(element);

        // 添加第一个节点
        if (root == null) {
            root = createNode(element, null);
            size++;

            // 新添加节点之后的处理
            afterAdd(root);
            return;
        }

        // 添加的不是第一个节点
        // 找到父节点
        Node<E> parent = root;
        Node<E> node = root;
        int cmp = 0;
        do {
            cmp = compare(element, node.element);
            parent = node;
            if (cmp > 0) {
                node = node.right;
            } else if (cmp < 0) {
                node = node.left;
            } else { // 相等
                node.element = element;
                return;
            }
        } while (node != null);

        // 看看插入到父节点的哪个位置
        Node<E> newNode = createNode(element, parent);
        if (cmp > 0) {
            parent.right = newNode;
        } else {
            parent.left = newNode;
        }
        size++;

        // 新添加节点之后的处理
        afterAdd(newNode);
    }

    /**
     * 添加node之后的调整
     *
     * @param node 新添加的节点
     */
    protected void afterAdd(Node<E> node) {
    }

    /**
     * 删除node之后的调整
     *
     * @param node 被删除的节点
     */
    protected void afterRemove(Node<E> node) {
    }

    public void remove(E element) {
        remove(node(element));
    }

    public boolean contains(E element) {
        return node(element) != null;
    }

    private void remove(Node<E> node) {
        if (node == null) return;

        size--;

        if (node.hasTwoChildren()) { // 度为2的节点
            // 找到后继节点
            Node<E> s = successor(node);
            // 用后继节点的值覆盖度为2的节点的值
            node.element = s.element;
            // 删除后继节点
            node = s;
        }

        // 删除node节点（node的度必然是1或者0）
        Node<E> replacement = node.left != null ? node.left : node.right;

        if (replacement != null) { // node是度为1的节点
            // 更改parent
            replacement.parent = node.parent;
            // 更改parent的left、right的指向
            if (node.parent == null) { // node是度为1的节点并且是根节点
                root = replacement;
            } else if (node == node.parent.left) {
                node.parent.left = replacement;
            } else { // node == node.parent.right
                node.parent.right = replacement;
            }

            // 删除节点之后的处理
            afterRemove(node);
        } else if (node.parent == null) { // node是叶子节点并且是根节点
            root = null;

            // 删除节点之后的处理
            afterRemove(node);
        } else { // node是叶子节点，但不是根节点
            if (node == node.parent.left) {
                node.parent.left = null;
            } else { // node == node.parent.right
                node.parent.right = null;
            }

            // 删除节点之后的处理
            afterRemove(node);
        }
    }

    private Node<E> node(E element) {
        Node<E> node = root;
        while (node != null) {
            int cmp = compare(element, node.element);
            if (cmp == 0) return node;
            if (cmp > 0) {
                node = node.right;
            } else { // cmp < 0
                node = node.left;
            }
        }
        return null;
    }

    /**
     * @return 返回值等于0，代表e1和e2相等；返回值大于0，代表e1大于e2；返回值小于于0，代表e1小于e2
     */
    private int compare(E e1, E e2) {
        if (comparator != null) {
            return comparator.compare(e1, e2);
        }
        return ((Comparable<E>) e1).compareTo(e2);
    }

    private void elementNotNullCheck(E element) {
        if (element == null) {
            throw new IllegalArgumentException("element must not be null");
        }
    }
}

  • BinaryTree

    public class BinaryTree<E> implements BinaryTreeInfo {
    protected int size;
    protected Node<E> root;

    public int size() {
        return size;
    }

    public boolean isEmpty() {
        return size == 0;
    }

    public void clear() {
        root = null;
        size = 0;
    }

    public void preorder(Visitor<E> visitor) {
        if (visitor == null) return;
        preorder(root, visitor);
    }

    private void preorder(Node<E> node, Visitor<E> visitor) {
        if (node == null || visitor.stop) return;

        visitor.stop = visitor.visit(node.element);
        preorder(node.left, visitor);
        preorder(node.right, visitor);
    }

    public void inorder(Visitor<E> visitor) {
        if (visitor == null) return;
        inorder(root, visitor);
    }

    private void inorder(Node<E> node, Visitor<E> visitor) {
        if (node == null || visitor.stop) return;

        inorder(node.left, visitor);
        if (visitor.stop) return;
        visitor.stop = visitor.visit(node.element);
        inorder(node.right, visitor);
    }

    public void postorder(Visitor<E> visitor) {
        if (visitor == null) return;
        postorder(root, visitor);
    }

    private void postorder(Node<E> node, Visitor<E> visitor) {
        if (node == null || visitor.stop) return;

        postorder(node.left, visitor);
        postorder(node.right, visitor);
        if (visitor.stop) return;
        visitor.stop = visitor.visit(node.element);
    }

    public void levelOrder(Visitor<E> visitor) {
        if (root == null || visitor == null) return;

        Queue<Node<E>> queue = new LinkedList<>();
        queue.offer(root);

        while (!queue.isEmpty()) {
            Node<E> node = queue.poll();
            if (visitor.visit(node.element)) return;

            if (node.left != null) {
                queue.offer(node.left);
            }

            if (node.right != null) {
                queue.offer(node.right);
            }
        }
    }

    public boolean isComplete() {
        if (root == null) return false;
        Queue<Node<E>> queue = new LinkedList<>();
        queue.offer(root);

        boolean leaf = false;
        while (!queue.isEmpty()) {
            Node<E> node = queue.poll();
            if (leaf && !node.isLeaf()) return false;

            if (node.left != null) {
                queue.offer(node.left);
            } else if (node.right != null) {
                return false;
            }

            if (node.right != null) {
                queue.offer(node.right);
            } else { // 后面遍历的节点都必须是叶子节点
                leaf = true;
            }
        }

        return true;
    }

    public int height() {
        if (root == null) return 0;

        // 树的高度
        int height = 0;
        // 存储着每一层的元素数量
        int levelSize = 1;
        Queue<Node<E>> queue = new LinkedList<>();
        queue.offer(root);

        while (!queue.isEmpty()) {
            Node<E> node = queue.poll();
            levelSize--;

            if (node.left != null) {
                queue.offer(node.left);
            }

            if (node.right != null) {
                queue.offer(node.right);
            }

            if (levelSize == 0) { // 意味着即将要访问下一层
                levelSize = queue.size();
                height++;
            }
        }

        return height;
    }

    public int height2() {
        return height(root);
    }

    private int height(Node<E> node) {
        if (node == null) return 0;
        return 1 + Math.max(height(node.left), height(node.right));
    }

    protected Node<E> createNode(E element, Node<E> parent) {
        return new Node<>(element, parent);
    }

    protected Node<E> predecessor(Node<E> node) {
        if (node == null) return null;

        // 前驱节点在左子树当中（left.right.right.right....）
        Node<E> p = node.left;
        if (p != null) {
            while (p.right != null) {
                p = p.right;
            }
            return p;
        }

        // 从父节点、祖父节点中寻找前驱节点
        while (node.parent != null && node == node.parent.left) {
            node = node.parent;
        }

        // node.parent == null
        // node == node.parent.right
        return node.parent;
    }

    protected Node<E> successor(Node<E> node) {
        if (node == null) return null;

        // 前驱节点在左子树当中（right.left.left.left....）
        Node<E> p = node.right;
        if (p != null) {
            while (p.left != null) {
                p = p.left;
            }
            return p;
        }

        // 从父节点、祖父节点中寻找前驱节点
        while (node.parent != null && node == node.parent.right) {
            node = node.parent;
        }

        return node.parent;
    }

    public static abstract class Visitor<E> {
        boolean stop;

        /**
         * @return 如果返回true，就代表停止遍历
         */
        abstract boolean visit(E element);
    }

    protected static class Node<E> {
        E element;
        Node<E> left;
        Node<E> right;
        Node<E> parent;

        public Node(E element, Node<E> parent) {
            this.element = element;
            this.parent = parent;
        }

        public boolean isLeaf() {
            return left == null && right == null;
        }

        public boolean hasTwoChildren() {
            return left != null && right != null;
        }

        public boolean isLeftChild() {
            return parent != null && this == parent.left;
        }

        public boolean isRightChild() {
            return parent != null && this == parent.right;
        }

        public Node<E> sibling() {
            if (isLeftChild()) {
                return parent.right;
            }

            if (isRightChild()) {
                return parent.left;
            }

            return null;
        }
    }

    @Override
    public Object root() {
        return root;
    }

    @Override
    public Object left(Object node) {
        return ((Node<E>) node).left;
    }

    @Override
    public Object right(Object node) {
        return ((Node<E>) node).right;
    }

    @Override
    public Object string(Object node) {
        return node;
    }
}

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