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二叉搜索树基本操作都可以在O(h)内完成，然而数的高度较高时，可能并不比在链表上执行的快。红黑树是平衡搜索树中的一种保证在最坏情况下时间复杂度为O(lg(n))
红黑树的性质
因为叶子节点是null，没有办法存储颜色，所以用一个哨兵nil，其他属性p、left、right、key是任意值(null)为好。
由红黑树的性质可知，h最大为2lg(n+1)
旋转操作
图片来源，其中关于红黑树的介绍更详细
LeftRotate(T, h) { x = h.right; //x的左子树成为h的右子树 h.right = x.left; if (x.left != T.nil) x.left.parent = h; //x的parent是h的parent x.parent = h.parent; //h的parent对应h的子树，成为x if (h.parent == T.nil) T.root = x; else if (h == h.parent.left) h.parent.left = x; else h.parent.right = x; //将x的左子树设置为h x.left = h; h.parent = y; }
RightRotate(T, x) { h = x.left; //h的右子树成为x的左子树 x.left = h.right; if (h.right != T.nil) h.right.parent = x; //h的parent是x的parent h.parent = x.parent; //x的parent对应x的子树，成为h if (x.parent == T.nil) T.root = h; else if (x == x.parent.right) x.parent.right = h; else x.parent.left = h; //将h的右子树设置为x h.right = x; x.parent = y; }
插入操作
插入操作与搜索二叉树操作基本相同。需要注意的是
TreeInsert(T, z) { y = T.nil; x = T.root; while (x != T.nil) { y = x; if (z.key < x.key) x = x.left; else x = x.right; } x.parent = y; if (y == T.nil) T.root = z; else if (z.key < y.key) y.left = z; else y.right = z; z.left = z.right = T.nil; z.color = RED; RbInsertFixup(T, z); }
理解RbInsertFixup
RbInserFixup(T, z) { while (z.parent.color == RED) { if (z.parent == z.parent.parent.left) { y = z.parent.parent.right; if (y.color == RED) { z.parent.color = BLACK; y.color = BLACK; z.parent.parent.color = RED; z = z.parent.parent; } else if (z == z.parent.right) { z = z.parent; LeftRotate(T, z); } z.parent.color = BLACK; z.parent.parent.color = RED; } else//将伤医情况的right与left换一下 { y = z.parent.parent.left; if (y.color == RED) { z.parent.color = BLACK; y.color = BLACK; z.parent.parent.color = RED; z = z.parent.parent; } else if (z == z.parent.left) { z = z.parent; LeftRotate(T, z); } z.parent.color = BLACK; z.parent.parent.color = RED; } } }
删除操作
情况与二叉搜索树类比。首先需要一个Transplant应用到红黑树上
RbTransplant(T, u, v) { if (u.parent == T.nil) T.root = v; else if (u == u.parent.left) u.parent.left = v; else u.parent.right = v; v.parent = u.parent; }
然后执行删除
RbTreeDelete(T, z) { y = z; yOriginalColor = y.color; if (z.left == T.nil) { x = z.right; RbTransplant(T, z, z.right); } else if (z.right == T.nil) { x = z.left; RbTransplant(T, z, z.left); } else { y = TreeMinimum(z.right);//z有双子，则后继为右侧最小 yOriginalColor = y.color; x = y.right; if (y.parent == z) { x.parent = y; } else { BrTransplant(T, y, y.right); y.right = z.right; y.right.parent = y; } BrTransplant(T, z, y); y.left = z.left; y.left.p = y; y.color = z.color; } if (yOriginalColor == BLACK) RbDeleteFixup(T, x); }
RbDeleteFixup(T, x) { while (x != T.root && x.color == BLACK) { if (x == x.parent.left) { w = x.parent.right; if (w.color == RED) { w.color = BLACK; x.p.color = RED; LeftRotate(T, x.parent); w = x.parent.right; } if (w.left.color == BLACK && w.right.color == BLACK) { w.color = RED; x = x.parent; } else { w.left.color = BLACK; w.color = RED; RightRotate(T, w); w.x.parent.right; } w.color = x.parent.color; x.parent.color = BLACK; w.right.color = BLACK; LeftRotate(T, x.p); x = T.root; } else //change right and left { w = x.parent.left; if (w.color == RED) { w.color = BLACK; x.p.color = RED; LeftRotate(T, x.parent); w = x.parent.left; } if (w.right.color == BLACK && w.left.color == BLACK) { w.color = RED; x = x.parent; } else { w.right.color = BLACK; w.color = RED; RightRotate(T, w); w.x.parent.left; } w.color = x.parent.color; x.parent.color = BLACK; w.left.color = BLACK; LeftRotate(T, x.p); x = T.root; } } x.color = BLACK; }
对红黑树的Extent
::动态顺序统计
在红黑树的每个节点中加入一个size属性，表示这棵树的节点数，其中哨兵的节点数size为0。则
x.size=x.left.size+z.right.size+1
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原文地址：http://www.cnblogs.com/qiusuo/p/5220720.html
