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利用Java实现红黑树

作者:胡不慌

红黑树是众多“平衡的”搜索树模式中的一种,在最坏情况下,它相关操作的时间复杂度为O(log n),接下俩小编将子啊下文详细介绍Java是如何实现红黑树的

1、红黑树的属性

红黑树是一种二分查找树,与普通的二分查找树不同的一点是,红黑树的每个节点都有一个颜色(color)属性。该属性的值要么是红色,要么是黑色。

通过限制从根到叶子的任何简单路径上的节点颜色,红黑树确保没有比任何其他路径长两倍的路径,从而使树近似平衡。

假设红黑树节点的属性有键(key)、颜色(color)、左子节点(left)、右子节点(right),父节点(parent)。

一棵红黑树必须满足下面有下面这些特性( 红黑树特性 ):

为了在红黑树代码中处理边界条件方便,我们用一个哨兵变量代替null。对于一个红黑树tree,哨兵变量RedBlackTree.NULL(下文代码中)是一个和其它节点有同样属性的节点,它的颜色(color)属性是黑色,其它属性可以任意取值。

我们使用哨兵变量是因为我们可以把一个节点node的子节点null当成一个普通节点。

在这里,我们使用哨兵变量RedBlackTree.NULL代替树中所有的null(所有的叶子节点及根节点的父节点)。

我们把从一个节点n(不包括)到任一叶子节点路径上的黑色节点的个数称为 黑色高度 ,用bh(n)表示。一棵红黑树的黑色高度是其根节点的黑色高度。

关于红黑树的搜索,求最小值,求最大值,求前驱,求后继这些操作的代码与二分查找树的这些操作的代码基本一致。可以在用java实现二分查找树查看。

结合上文给出下面的代码。

用一个枚举类Color表示颜色:

public enum Color {
    Black("黑色"), Red("红色");

    private String color;

    private Color(String color) {
        this.color = color;
    }

    @Override
    public String toString() {
        return color;
    }
}

类Node表示节点:

public class Node {
    public int key;
    public Color color;
    public Node left;
    public Node right;
    public Node parent;

    public Node() {
    }

    public Node(Color color) {
        this.color = color;
    }

    public Node(int key) {
        this.key = key;
        this.color = Color.Red;
    }

    public int height() {
        return Math.max(left != RedBlackTree.NULL ? left.height() : 0, right != RedBlackTree.NULL ? right.height() : 0) + 1;
    }

    public Node minimum() {
        Node pointer = this;
        while (pointer.left != RedBlackTree.NULL)
            pointer = pointer.left;
        return pointer;
    }

    @Override
    public String toString() {
        String position = "null";
        if (this.parent != RedBlackTree.NULL)
            position = this.parent.left == this ? "left" : "right";
        return "[key: " + key + ", color: " + color + ", parent: " + parent.key + ", position: " + position + "]";
    }
}

类RedTreeNode表示红黑树:

public class RedBlackTree {

    // 表示哨兵变量
    public final static Node NULL = new Node(Color.Black);

    public Node root;

    public RedBlackTree() {
        this.root = NULL;
    }

}

2、旋转

红黑树的插入和删除操作,能改变红黑树的结构,可能会使它不再有前面所说的某些特性性。为了维持这些特性,我们需要改变树中某些节点的颜色和位置。

我们可以通过旋转改变节点的结构。主要有 左旋转 右旋转 两种方式。具体如下图所示。

左旋转:把一个节点n的右子节点right变为它的父节点,n变为right的左子节点,所以right不能为null。这时n的右指针空了出来,right的左子树被n挤掉,所以right原来的左子树称为n的右子树。

右旋转:把一个节点n的左子节点left变为它的父节点,n变为left的右子节点,所以left不能为null。这时n的左指针被空了出来,left的右子树被n挤掉,所以left原来的右子树被称为n的左子树。

可在RedTreeNode类中,加上如下实现代码:

public void leftRotate(Node node) {
        Node rightNode = node.right;

        node.right = rightNode.left;
        if (rightNode.left != RedBlackTree.NULL)
            rightNode.left.parent = node;

        rightNode.parent = node.parent;
        if (node.parent == RedBlackTree.NULL)
            this.root = rightNode;
        else if (node.parent.left == node)
            node.parent.left = rightNode;
        else
            node.parent.right = rightNode;

        rightNode.left = node;
        node.parent = rightNode;
    }

    public void rightRotate(Node node) {
        Node leftNode = node.left;

        node.left = leftNode.right;
        if (leftNode.right != RedBlackTree.NULL)
            leftNode.right.parent = node;

        leftNode.parent = node.parent;
        if (node.parent == RedBlackTree.NULL) {
            this.root = leftNode;
        } else if (node.parent.left == node) {
            node.parent.left = leftNode;
        } else {
            node.parent.right = leftNode;
        }

        leftNode.right = node;
        node.parent = leftNode;
    }

3、插入

红黑树的插入代码与二分查找树的插入代码非常相似。只不过红黑树的插入操作会改变红黑树的结构,使其不在有该有的特性。

在这里,新插入的节点默认是红色。

所以在插入节点之后,要有维护红黑树特性的代码。

public void insert(Node node) {
        Node parentPointer = RedBlackTree.NULL;
        Node pointer = this.root;

        while (this.root != RedBlackTree.NULL) {
            parentPointer = pointer;
            pointer = node.key < pointer.key ? pointer.left : pointer.right;
        }

        node.parent = parentPointer;
        if(parentPointer == RedBlackTree.NULL) {
            this.root = node;
        }else if(node.key < parentPointer.key) {
            parentPointer.left = node;
        }else {
            parentPointer.right = node;
        }

        node.left = RedBlackTree.NULL;
        node.right = RedBlackTree.NULL;
        node.color = Color.Red;
        // 维护红黑树属性的方法
        this.insertFixUp(node);
    }

用上述方法插入一个新节点的时候,有两类情况会违反红黑树的特性。

对于第一类情况,可以直接把根结点设置为黑色;而针对第二类情况,需要根据具体条件,做出相应的解决方案。

具体代码如下:

public void insertFixUp(Node node) {
        // 当node不是根结点,且node的父节点颜色为红色
        while (node.parent.color == Color.Red) {
            // 先判断node的父节点是左子节点,还是右子节点,这不同的情况,解决方案也会不同
            if (node.parent == node.parent.parent.left) {
                Node uncleNode = node.parent.parent.right;
                if (uncleNode.color == Color.Red) {  // 如果叔叔节点是红色,则父父一定是黑色
                    // 通过把父父节点变成红色,父节点和兄弟节点变成黑色,然后在判断父父节点的颜色是否合适
                    uncleNode.color = Color.Black;
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    node = node.parent.parent;
                } else if (node == node.parent.right) {
                    node = node.parent;
                    this.leftRotate(node);
                } else {
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    this.rightRotate(node.parent.parent);
                }
            } else {
                Node uncleNode = node.parent.parent.left;
                if (uncleNode.color == Color.Red) {
                    uncleNode.color = Color.Black;
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    node = node.parent.parent;
                } else if (node == node.parent.left) {
                    node = node.parent;
                    this.rightRotate(node);
                } else {
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    this.leftRotate(node.parent.parent);
                }
            }
        }
        // 如果之前树中没有节点,那么新加入的点就成了新结点,而新插入的结点都是红色的,所以需要修改。
        this.root.color = Color.Black;
    }

下面的图分别对应第二类情况中的六种及相应处理结果。

情况1:

情况2:

情况3:

情况4:

情况5:

情况6:

4、删除

红黑树中节点的删除会使一个结点代替另外一个节点。所以先要实现这样的代码:

public void transplant(Node n1, Node n2) {
        if(n1.parent == RedBlackTree.NULL){
            this.root = n2;
        }else if(n1.parent.left == n1) {
            n1.parent.left = n2;
        }else {
            n1.parent.right = n2;
        }
        n2.parent = n1.parent;
    }


红黑树的删除节点代码是基于二分查找树的删除节点代码而写的。

删除结点代码:

public void delete(Node node) {
        Node pointer1 = node;
        // 用于记录被删除的颜色,如果是红色,可以不用管,但如果是黑色,可能会破坏红黑树的属性
        Color pointerOriginColor = pointer1.color;
        // 用于记录问题的出现点
        Node pointer2;
        if (node.left == RedBlackTree.NULL) {
            pointer2 = node.right;
            this.transplant(node, node.right);
        } else if (node.right == RedBlackTree.NULL) {
            pointer2 = node.left;
            this.transplant(node, node.left);
        } else {
            // 如要删除的字节有两个子节点,则找到其直接后继(右子树最小值),直接后继节点没有非空左子节点。
            pointer1 = node.right.minimum();
            // 记录直接后继的颜色和其右子节点
            pointerOriginColor = pointer1.color;
            pointer2 = pointer1.right;
            // 如果其直接后继是node的右子节点,不用进行处理
            if (pointer1.parent == node) {
                pointer2.parent = pointer1;
            } else {
                // 否则,先把直接后继从树中提取出来,用来替换node
                this.transplant(pointer1, pointer1.right);
                pointer1.right = node.right;
                pointer1.right.parent = pointer1;
            }
            // 用node的直接后继替换node,继承node的颜色
            this.transplant(node, pointer1);
            pointer1.left = node.left;
            pointer1.left.parent = pointer1;
            pointer1.color = node.color;
        }
        if (pointerOriginColor == Color.Black) {
            this.deleteFixUp(pointer2);
        }
    }

当被删除节点的颜色是黑色时需要调用方法维护红黑树的特性。

主要有两类情况:

private void deleteFixUp(Node node) {
        // 如果node不是根节点,且是黑色
        while (node != this.root && node.color == Color.Black) {
            // 如果node是其父节点的左子节点
            if (node == node.parent.left) {
                // 记录node的兄弟节点
                Node pointer1 = node.parent.right;
                // 如果他兄弟节点是红色
                if (pointer1.color == Color.Red) {
                    pointer1.color = Color.Black;
                    node.parent.color = Color.Red;
                    leftRotate(node.parent);
                    pointer1 = node.parent.right;
                }
                if (pointer1.left.color == Color.Black && pointer1.right.color == Color.Black) {
                    pointer1.color = Color.Red;
                    node = node.parent;
                } else if (pointer1.right.color == Color.Black) {
                    pointer1.left.color = Color.Black;
                    pointer1.color = Color.Red;
                    rightRotate(pointer1);
                    pointer1 = node.parent.right;
                } else {
                    pointer1.color = node.parent.color;
                    node.parent.color = Color.Black;
                    pointer1.right.color = Color.Black;
                    leftRotate(node.parent);
                    node = this.root;
                }
            } else {
                // 记录node的兄弟节点
                Node pointer1 = node.parent.left;
                // 如果他兄弟节点是红色
                if (pointer1.color == Color.Red) {
                    pointer1.color = Color.Black;
                    node.parent.color = Color.Red;
                    rightRotate(node.parent);
                    pointer1 = node.parent.left;
                }
                if (pointer1.right.color == Color.Black && pointer1.left.color == Color.Black) {
                    pointer1.color = Color.Red;
                    node = node.parent;
                } else if (pointer1.left.color == Color.Black) {
                    pointer1.right.color = Color.Black;
                    pointer1.color = Color.Red;
                    leftRotate(pointer1);
                    pointer1 = node.parent.left;
                } else {
                    pointer1.color = node.parent.color;
                    node.parent.color = Color.Black;
                    pointer1.left.color = Color.Black;
                    rightRotate(node.parent);
                    node = this.root;
                }
            }

        }
        node.color = Color.Black;
    }

对第二类情况,有下面8种:

情况1:

情况2:

情况3:

情况4:

情况5:

情况6:

情况7:

情况8:

5、所有代码

public enum Color {
    Black("黑色"), Red("红色");

    private String color;

    private Color(String color) {
        this.color = color;
    }

    @Override
    public String toString() {
        return color;
    }
}
public class Node {
    public int key;
    public Color color;
    public Node left;
    public Node right;
    public Node parent;

    public Node() {
    }

    public Node(Color color) {
        this.color = color;
    }

    public Node(int key) {
        this.key = key;
        this.color = Color.Red;
    }

    /**
     * 求在树中节点的高度
     * 
     * @return
     */
    public int height() {
        return Math.max(left != RedBlackTree.NULL ? left.height() : 0, right != RedBlackTree.NULL ? right.height() : 0) + 1;
    }

    /**
     * 在以该节点为根节点的树中,求最小节点
     * 
     * @return
     */
    public Node minimum() {
        Node pointer = this;
        while (pointer.left != RedBlackTree.NULL)
            pointer = pointer.left;
        return pointer;
    }

    @Override
    public String toString() {
        String position = "null";
        if (this.parent != RedBlackTree.NULL)
            position = this.parent.left == this ? "left" : "right";
        return "[key: " + key + ", color: " + color + ", parent: " + parent.key + ", position: " + position + "]";
    }
}
import java.util.LinkedList;
import java.util.Queue;

public class RedBlackTree {

    public final static Node NULL = new Node(Color.Black);

    public Node root;

    public RedBlackTree() {
        this.root = NULL;
    }

    /**
     * 左旋转
     * 
     * @param node
     */
    public void leftRotate(Node node) {
        Node rightNode = node.right;

        node.right = rightNode.left;
        if (rightNode.left != RedBlackTree.NULL)
            rightNode.left.parent = node;

        rightNode.parent = node.parent;
        if (node.parent == RedBlackTree.NULL)
            this.root = rightNode;
        else if (node.parent.left == node)
            node.parent.left = rightNode;
        else
            node.parent.right = rightNode;

        rightNode.left = node;
        node.parent = rightNode;
    }

    /**
     * 右旋转
     * 
     * @param node
     */
    public void rightRotate(Node node) {
        Node leftNode = node.left;

        node.left = leftNode.right;
        if (leftNode.right != RedBlackTree.NULL)
            leftNode.right.parent = node;

        leftNode.parent = node.parent;
        if (node.parent == RedBlackTree.NULL) {
            this.root = leftNode;
        } else if (node.parent.left == node) {
            node.parent.left = leftNode;
        } else {
            node.parent.right = leftNode;
        }

        leftNode.right = node;
        node.parent = leftNode;
    }

    public void insert(Node node) {
        Node parentPointer = RedBlackTree.NULL;
        Node pointer = this.root;

        while (pointer != RedBlackTree.NULL) {
            parentPointer = pointer;
            pointer = node.key < pointer.key ? pointer.left : pointer.right;
        }

        node.parent = parentPointer;
        if (parentPointer == RedBlackTree.NULL) {
            this.root = node;
        } else if (node.key < parentPointer.key) {
            parentPointer.left = node;
        } else {
            parentPointer.right = node;
        }

        node.left = RedBlackTree.NULL;
        node.right = RedBlackTree.NULL;
        node.color = Color.Red;
        this.insertFixUp(node);
    }

    private void insertFixUp(Node node) {
        // 当node不是根结点,且node的父节点颜色为红色
        while (node.parent.color == Color.Red) {
            // 先判断node的父节点是左子节点,还是右子节点,这不同的情况,解决方案也会不同
            if (node.parent == node.parent.parent.left) {
                Node uncleNode = node.parent.parent.right;
                if (uncleNode.color == Color.Red) { // 如果叔叔节点是红色,则父父一定是黑色
                    // 通过把父父节点变成红色,父节点和兄弟节点变成黑色,然后在判断父父节点的颜色是否合适
                    uncleNode.color = Color.Black;
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    node = node.parent.parent;
                } else if (node == node.parent.right) { // node是其父节点的右子节点,且叔叔节点是黑色
                    // 对node的父节点进行左旋转
                    node = node.parent;
                    this.leftRotate(node);
                } else { // node如果叔叔节点是黑色,node是其父节点的左子节点,父父节点是黑色
                    // 把父节点变成黑色,父父节点变成红色,对父父节点进行右旋转
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    this.rightRotate(node.parent.parent);
                }
            } else {
                Node uncleNode = node.parent.parent.left;
                if (uncleNode.color == Color.Red) {
                    uncleNode.color = Color.Black;
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    node = node.parent.parent;
                } else if (node == node.parent.left) {
                    node = node.parent;
                    this.rightRotate(node);
                } else {
                    node.parent.color = Color.Black;
                    node.parent.parent.color = Color.Red;
                    this.leftRotate(node.parent.parent);
                }
            }
        }
        // 如果之前树中没有节点,那么新加入的点就成了新结点,而新插入的结点都是红色的,所以需要修改。
        this.root.color = Color.Black;
    }

    /**
     * n2替代n1
     * 
     * @param n1
     * @param n2
     */
    private void transplant(Node n1, Node n2) {

        if (n1.parent == RedBlackTree.NULL) { // 如果n1是根节点
            this.root = n2;
        } else if (n1.parent.left == n1) { // 如果n1是其父节点的左子节点
            n1.parent.left = n2;
        } else { // 如果n1是其父节点的右子节点
            n1.parent.right = n2;
        }
        n2.parent = n1.parent;
    }

    /**
     * 删除节点node
     * 
     * @param node
     */
    public void delete(Node node) {
        Node pointer1 = node;
        // 用于记录被删除的颜色,如果是红色,可以不用管,但如果是黑色,可能会破坏红黑树的属性
        Color pointerOriginColor = pointer1.color;
        // 用于记录问题的出现点
        Node pointer2;
        if (node.left == RedBlackTree.NULL) {
            pointer2 = node.right;
            this.transplant(node, node.right);
        } else if (node.right == RedBlackTree.NULL) {
            pointer2 = node.left;
            this.transplant(node, node.left);
        } else {
            // 如要删除的字节有两个子节点,则找到其直接后继(右子树最小值),直接后继节点没有非空左子节点。
            pointer1 = node.right.minimum();
            // 记录直接后继的颜色和其右子节点
            pointerOriginColor = pointer1.color;
            pointer2 = pointer1.right;
            // 如果其直接后继是node的右子节点,不用进行处理
            if (pointer1.parent == node) {
                pointer2.parent = pointer1;
            } else {
                // 否则,先把直接后继从树中提取出来,用来替换node
                this.transplant(pointer1, pointer1.right);
                pointer1.right = node.right;
                pointer1.right.parent = pointer1;
            }
            // 用node的直接后继替换node,继承node的颜色
            this.transplant(node, pointer1);
            pointer1.left = node.left;
            pointer1.left.parent = pointer1;
            pointer1.color = node.color;
        }
        if (pointerOriginColor == Color.Black) {
            this.deleteFixUp(pointer2);
        }
    }

    /**
     * The procedure RB-DELETE-FIXUP restores properties 1, 2, and 4
     * 
     * @param node
     */
    private void deleteFixUp(Node node) {
        // 如果node不是根节点,且是黑色
        while (node != this.root && node.color == Color.Black) {
            // 如果node是其父节点的左子节点
            if (node == node.parent.left) {
                // 记录node的兄弟节点
                Node pointer1 = node.parent.right;
                // 如果node兄弟节点是红色
                if (pointer1.color == Color.Red) {
                    pointer1.color = Color.Black;
                    node.parent.color = Color.Red;
                    leftRotate(node.parent);
                    pointer1 = node.parent.right;
                }
                if (pointer1.left.color == Color.Black && pointer1.right.color == Color.Black) {
                    pointer1.color = Color.Red;
                    node = node.parent;
                } else if (pointer1.right.color == Color.Black) {
                    pointer1.left.color = Color.Black;
                    pointer1.color = Color.Red;
                    rightRotate(pointer1);
                    pointer1 = node.parent.right;
                } else {
                    pointer1.color = node.parent.color;
                    node.parent.color = Color.Black;
                    pointer1.right.color = Color.Black;
                    leftRotate(node.parent);
                    node = this.root;
                }
            } else {
                // 记录node的兄弟节点
                Node pointer1 = node.parent.left;
                // 如果他兄弟节点是红色
                if (pointer1.color == Color.Red) {
                    pointer1.color = Color.Black;
                    node.parent.color = Color.Red;
                    rightRotate(node.parent);
                    pointer1 = node.parent.left;
                }
                if (pointer1.right.color == Color.Black && pointer1.left.color == Color.Black) {
                    pointer1.color = Color.Red;
                    node = node.parent;
                } else if (pointer1.left.color == Color.Black) {
                    pointer1.right.color = Color.Black;
                    pointer1.color = Color.Red;
                    leftRotate(pointer1);
                    pointer1 = node.parent.left;
                } else {
                    pointer1.color = node.parent.color;
                    node.parent.color = Color.Black;
                    pointer1.left.color = Color.Black;
                    rightRotate(node.parent);
                    node = this.root;
                }
            }

        }
        node.color = Color.Black;
    }

    private void innerWalk(Node node) {
        if (node != NULL) {
            innerWalk(node.left);
            System.out.println(node);
            innerWalk(node.right);
        }
    }

    /**
     * 中序遍历
     */
    public void innerWalk() {
        this.innerWalk(this.root);
    }

    /**
     * 层次遍历
     */
    public void print() {
        Queue<Node> queue = new LinkedList<>();
        queue.add(this.root);
        while (!queue.isEmpty()) {
            Node temp = queue.poll();
            System.out.println(temp);
            if (temp.left != NULL)
                queue.add(temp.left);
            if (temp.right != NULL)
                queue.add(temp.right);
        }
    }

    // 查找
    public Node search(int key) {
        Node pointer = this.root;
        while (pointer != NULL && pointer.key != key) {
            pointer = pointer.key < key ? pointer.right : pointer.left;
        }
        return pointer;
    }

}

6、演示

演示代码:

public class Test01 {
  public static void main(String[] args) {
    int[] arr = { 1, 2, 3, 4, 5, 6, 7, 8 };
    RedBlackTree redBlackTree = new RedBlackTree();
    for (int i = 0; i < arr.length; i++) {
      redBlackTree.insert(new Node(arr[i]));
    }
    System.out.println("树的高度: " + redBlackTree.root.height());
    System.out.println("左子树的高度: " + redBlackTree.root.left.height());
    System.out.println("右子树的高度: " + redBlackTree.root.right.height());
    System.out.println("层次遍历");
    redBlackTree.print();
    // 要删除节点
    Node node = redBlackTree.search(4);
    redBlackTree.delete(node);
    System.out.println("树的高度: " + redBlackTree.root.height());
    System.out.println("左子树的高度: " + redBlackTree.root.left.height());
    System.out.println("右子树的高度: " + redBlackTree.root.right.height());
    System.out.println("层次遍历");
    redBlackTree.print();
  }
}

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