什么是红黑树？
红黑树是特殊的二叉查找树，意味着它满足二叉查找树的特征：
若任意节点的左子树不空，则左子树上所有节点的值均小于它的根节点的值；
若任意节点的右子树不空，则右子树上所有节点的值均大于它的根节点的值；
任意节点的左、右子树也分别为二叉查找树；
没有键值相等的节点。
有哪些性质？
（1）每个节点或者是黑色，或者是红色。
（2）根节点是黑色。
（3）每个叶子节点（NIL）是黑色。 [注意：这里叶子节点，是指为空(NIL或NULL)的叶子节点！]
（4）如果一个节点是红色的，则它的子节点必须是黑色的。
（5）从一个节点到该节点的子孙节点的所有路径上包含相同数目的黑节点。（不理解啊）
注意：
(01) 特性(3)中的叶子节点，是只为空(NIL或null)的节点。
(02) 特性(5)，确保没有一条路径会比其他路径长出俩倍。因而，红黑树是相对是接近平衡的二叉树。
为什么要用它？
主要是用它来存储有序的数据，它的时间复杂度是O(lgn)，效率非常之高。
有哪些操作？
插入
1、插入
2、着色（红色）
3、旋转并重新着色
（1）被插入节点是根节点（第一次插入）
-》直接把此节点涂为黑色
（2）被插入的节点的父节点是黑色
-》什么也不需要做。节点被插入后，仍然是红黑树
（3）被插入的节点的父节点是红色，当前节点的父节点是当前节点的祖父节点的左孩子
（3-1）叔叔节点也是红色
-》将“父节点”设为黑色。
-》将“叔叔节点”设为黑色。
-》 将“祖父节点”设为“红色”。
-》将“祖父节点”设为“当前节点”(红色节点)；即，之后继续对“当前节点”进行操作。
（3-2）叔叔节点是黑色，且当前节点是其父节点的右孩子
-》将“父节点”作为“新的当前节点”
-》以“新的当前节点”为支点进行左旋
（3-3）叔叔节点是黑色，且当前节点是其父节点的左孩子
-》将“父节点”设为“黑色”。
-》将“祖父节点”设为“红色”。
-》以“祖父节点”为支点进行右旋。
（3）被插入的节点的父节点是红色，当前节点的父节点是当前节点的祖父节点的右孩子
（3-1）叔叔节点也是红色
-》将“父节点”设为黑色。
-》将“叔叔节点”设为黑色。
-》 将“祖父节点”设为“红色”。
-》将“祖父节点”设为“当前节点”(红色节点)；即，之后继续对“当前节点”进行操作。
（3-2）叔叔节点是黑色，且当前节点是其父节点的右孩子
-》将“父节点”设为“黑色”
-》将“祖父节点”设为“红色”
-》以“祖父节点”为支点进行左旋
（3-3）叔叔节点是黑色，且当前节点是其父节点的左孩子
-》将“父节点”设为“黑色”。
-》将“祖父节点”设为“红色”。
-》以“祖父节点”为支点进行左旋。
删除
1、将红黑树当作一颗二叉查找树，将节点删除
（1-1）被删除节点没有儿子，即为叶节点
-》直接将该节点删除就OK了
（1-2）被删除节点只有一个儿子
-》直接删除该节点，并用该节点的唯一子节点顶替它的位置
（1-3）被删除节点有两个儿子
-》先找出它的后继节点；
-》然后把“它的后继节点的内容”复制给“该节点的内容”；
-》之后删除“它的后继节点”
2、通过"旋转和重新着色"等一系列来修正该树，使之重新成为一棵红黑树
查找
旋转
左旋
以A节点左旋，假设A的右孩子为B，
A变为B的左孩子，B的左孩子变为A的右孩子，如果A的parent不为空，A的parent变为B的parent
右旋
以A节点右旋，假设A的左孩子为B，
A变为B的右孩子，B的右孩子变为A的左孩子，如果A的parent不为空，A的parent变为B的parent
Java中哪里实现了红黑树？
HashMap
TreeMap
Java8HashMap中红黑树实现的源码如下：
interface Entry<K,V> {
K getKey();
V getValue();
V setValue(V value);
boolean equals(Object o);
int hashCode();
public static <K extends Comparable<? super K>, V> Comparator<Map.Entry<K,V>> comparingByKey() {
return (Comparator<Map.Entry<K, V>> & Serializable)
(c1, c2) -> c1.getKey().compareTo(c2.getKey());
}
public static <K, V extends Comparable<? super V>> Comparator<Map.Entry<K,V>> comparingByValue() {
return (Comparator<Map.Entry<K, V>> & Serializable)
(c1, c2) -> c1.getValue().compareTo(c2.getValue());
}
public static <K, V> Comparator<Map.Entry<K, V>> comparingByKey(Comparator<? super K> cmp) {
Objects.requireNonNull(cmp);
return (Comparator<Map.Entry<K, V>> & Serializable)
(c1, c2) -> cmp.compare(c1.getKey(), c2.getKey());
}
public static <K, V> Comparator<Map.Entry<K, V>> comparingByValue(Comparator<? super V> cmp) {
Objects.requireNonNull(cmp);
return (Comparator<Map.Entry<K, V>> & Serializable)
(c1, c2) -> cmp.compare(c1.getValue(), c2.getValue());
}
}
static class Node<K,V> implements Map.Entry<K,V> {
final int hash;
final K key;
V value;
Node<K,V> next;
Node(int hash, K key, V value, Node<K,V> next) {
this.hash = hash;
this.key = key;
this.value = value;
this.next = next;
}
public final K getKey() { return key; }
public final V getValue() { return value; }
public final String toString() { return key + "=" + value; }
public final int hashCode() {
return Objects.hashCode(key) ^ Objects.hashCode(value);
}
public final V setValue(V newValue) {
V oldValue = value;
value = newValue;
return oldValue;
}
public final boolean equals(Object o) {
if (o == this)
return true;
if (o instanceof Map.Entry) {
Map.Entry<?,?> e = (Map.Entry<?,?>)o;
if (Objects.equals(key, e.getKey()) &&
Objects.equals(value, e.getValue()))
return true;
}
return false;
}
}
static class Entry<K,V> extends HashMap.Node<K,V> {
Entry<K,V> before, after;
Entry(int hash, K key, V value, Node<K,V> next) {
super(hash, key, value, next);
}
}
//节点的结构定义TreeNode，先得搞清楚next、prev、parent、left、right是指什么？
static final class TreeNode<K,V> extends LinkedHashMap.Entry<K,V> {
TreeNode<K,V> parent; // 父节点
TreeNode<K,V> left; //左孩子节点
TreeNode<K,V> right; //右孩子节点
TreeNode<K,V> prev; //
boolean red; //节点的颜色
TreeNode(int hash, K key, V val, Node<K,V> next) {
super(hash, key, val, next);
}
//获取根节点
final TreeNode<K,V> root() {
for (TreeNode<K,V> r = this, p;;) {
if ((p = r.parent) == null)
return r;
r = p;
}
}
/**
* 将root转为根节点
*/
static <K,V> void moveRootToFront(Node<K,V>[] tab, TreeNode<K,V> root) {
int n;
if (root != null && tab != null && (n = tab.length) > 0) {
int index = (n - 1) & root.hash;//计算一个索引，为什么这么计算稍后研究
TreeNode<K,V> first = (TreeNode<K,V>)tab[index];
if (root != first) {
Node<K,V> rn;
tab[index] = root;//root替换first
TreeNode<K,V> rp = root.prev;
if ((rn = root.next) != null)
((TreeNode<K,V>)rn).prev = rp;
if (rp != null)
rp.next = rn;
if (first != null)
first.prev = root;
root.next = first;
root.prev = null;
}
assert checkInvariants(root);//断言，检查不变量，无需考虑
}
}
/**
* 根据hash和key找节点
*/
final TreeNode<K,V> find(int h, Object k, Class<?> kc) {
TreeNode<K,V> p = this;
do {
int ph, dir; K pk;
TreeNode<K,V> pl = p.left, pr = p.right, q;
//先判断hash，大于p的hash，将p.right赋给p，往右查找，否则往左查找
if ((ph = p.hash) > h)
p = pl;
else if (ph < h)
p = pr;
else if ((pk = p.key) == k || (k != null && k.equals(pk)))//key相等就找到，返回
return p;
else if (pl == null)//如果左节点为空，往右走
p = pr;
else if (pr == null)//如果右节点为空，往左走
p = pl;
else if ((kc != null ||
(kc = comparableClassFor(k)) != null) &&
(dir = compareComparables(kc, k, pk)) != 0)//如果kc实现了Comparable并且k小于pk，往左走，否则往右
p = (dir < 0) ? pl : pr;
else if ((q = pr.find(h, k, kc)) != null)
return q;
else
p = pl;
} while (p != null);
return null;
}
/**
* 从根节点开始查找
*/
final TreeNode<K,V> getTreeNode(int h, Object k) {
return ((parent != null) ? root() : this).find(h, k, null);
}
/**
* 比较a和b的大小
*/
static int tieBreakOrder(Object a, Object b) {
int d;
if (a == null || b == null ||
(d = a.getClass().getName().
compareTo(b.getClass().getName())) == 0)
d = (System.identityHashCode(a) <= System.identityHashCode(b) ?
-1 : 1);
return d;
}
/**
* 链表转为红黑树
*/
final void treeify(Node<K,V>[] tab) {
TreeNode<K,V> root = null;
for (TreeNode<K,V> x = this, next; x != null; x = next) {
next = (TreeNode<K,V>)x.next;
x.left = x.right = null;
if (root == null) {
x.parent = null;
x.red = false;
root = x;
}
else {
K k = x.key;
int h = x.hash;
Class<?> kc = null;
for (TreeNode<K,V> p = root;;) {
int dir, ph;
K pk = p.key;
if ((ph = p.hash) > h)
dir = -1;
else if (ph < h)
dir = 1;
else if ((kc == null &&
(kc = comparableClassFor(k)) == null) ||
(dir = compareComparables(kc, k, pk)) == 0)
dir = tieBreakOrder(k, pk);
TreeNode<K,V> xp = p;
if ((p = (dir <= 0) ? p.left : p.right) == null) {
x.parent = xp;
if (dir <= 0)
xp.left = x;
else
xp.right = x;
root = balanceInsertion(root, x);//插入后进行旋转平衡，稍后研究改方法
break;
}
}
}
}
moveRootToFront(tab, root);//将root转为根节点
}
/**
* 将红黑树节点转为链表节点
*/
final Node<K,V> untreeify(HashMap<K,V> map) {
Node<K,V> hd = null, tl = null;
for (Node<K,V> q = this; q != null; q = q.next) {
Node<K,V> p = map.replacementNode(q, null);//根据当前节点创建了一个新的Node，next为null
if (tl == null)
hd = p;
else
tl.next = p;
tl = p;
}
return hd;
}
/**
* 红黑树节点的插入
*/
final TreeNode<K,V> putTreeVal(HashMap<K,V> map, Node<K,V>[] tab,
int h, K k, V v) {
Class<?> kc = null;
boolean searched = false;
TreeNode<K,V> root = (parent != null) ? root() : this;//找root节点
for (TreeNode<K,V> p = root;;) {
//先找，找到就返回
int dir, ph; K pk;
if ((ph = p.hash) > h)
dir = -1;
else if (ph < h)
dir = 1;
else if ((pk = p.key) == k || (k != null && k.equals(pk)))
return p;
else if ((kc == null &&
(kc = comparableClassFor(k)) == null) ||
(dir = compareComparables(kc, k, pk)) == 0) {
if (!searched) {
TreeNode<K,V> q, ch;
searched = true;
if (((ch = p.left) != null &&
(q = ch.find(h, k, kc)) != null) ||
((ch = p.right) != null &&
(q = ch.find(h, k, kc)) != null))
return q;
}
dir = tieBreakOrder(k, pk);
}
//没找到，根据之前的dir确认往哪边插入
TreeNode<K,V> xp = p;
if ((p = (dir <= 0) ? p.left : p.right) == null) {
Node<K,V> xpn = xp.next;
TreeNode<K,V> x = map.newTreeNode(h, k, v, xpn);
if (dir <= 0)
xp.left = x;
else
xp.right = x;
xp.next = x;
x.parent = x.prev = xp;
if (xpn != null)
((TreeNode<K,V>)xpn).prev = x;
moveRootToFront(tab, balanceInsertion(root, x));//平衡操作
return null;
}
}
}
/**
* 红黑树节点的删除（删除调用者）
*/
final void removeTreeNode(HashMap<K,V> map, Node<K,V>[] tab,
boolean movable) {
int n;
if (tab == null || (n = tab.length) == 0)
return;
int index = (n - 1) & hash;
TreeNode<K,V> first = (TreeNode<K,V>)tab[index], root = first, rl;
TreeNode<K,V> succ = (TreeNode<K,V>)next, pred = prev;
if (pred == null)
tab[index] = first = succ;
else
pred.next = succ;
if (succ != null)
succ.prev = pred;
if (first == null)
return;
if (root.parent != null)
root = root.root();
if (root == null || root.right == null ||
(rl = root.left) == null || rl.left == null) {
tab[index] = first.untreeify(map); // 删除之后节点太少，转为链表
return;
}
TreeNode<K,V> p = this, pl = left, pr = right, replacement;
if (pl != null && pr != null) {
TreeNode<K,V> s = pr, sl;
while ((sl = s.left) != null) // find successor
s = sl;
boolean c = s.red; s.red = p.red; p.red = c; // swap colors
TreeNode<K,V> sr = s.right;
TreeNode<K,V> pp = p.parent;
if (s == pr) { // p was s's direct parent
p.parent = s;
s.right = p;
}
else {
TreeNode<K,V> sp = s.parent;
if ((p.parent = sp) != null) {
if (s == sp.left)
sp.left = p;
else
sp.right = p;
}
if ((s.right = pr) != null)
pr.parent = s;
}
p.left = null;
if ((p.right = sr) != null)
sr.parent = p;
if ((s.left = pl) != null)
pl.parent = s;
if ((s.parent = pp) == null)
root = s;
else if (p == pp.left)
pp.left = s;
else
pp.right = s;
if (sr != null)
replacement = sr;
else
replacement = p;
}
else if (pl != null)
replacement = pl;
else if (pr != null)
replacement = pr;
else
replacement = p;
if (replacement != p) {
TreeNode<K,V> pp = replacement.parent = p.parent;
if (pp == null)
root = replacement;
else if (p == pp.left)
pp.left = replacement;
else
pp.right = replacement;
p.left = p.right = p.parent = null;
}
TreeNode<K,V> r = p.red ? root : balanceDeletion(root, replacement);
if (replacement == p) { // detach
TreeNode<K,V> pp = p.parent;
p.parent = null;
if (pp != null) {
if (p == pp.left)
pp.left = null;
else if (p == pp.right)
pp.right = null;
}
}
if (movable)
moveRootToFront(tab, r);
}
/**
*
*/
final void split(HashMap<K,V> map, Node<K,V>[] tab, int index, int bit) {
TreeNode<K,V> b = this;
// Relink into lo and hi lists, preserving order
TreeNode<K,V> loHead = null, loTail = null;
TreeNode<K,V> hiHead = null, hiTail = null;
int lc = 0, hc = 0;
for (TreeNode<K,V> e = b, next; e != null; e = next) {
next = (TreeNode<K,V>)e.next;
e.next = null;
if ((e.hash & bit) == 0) {
if ((e.prev = loTail) == null)
loHead = e;
else
loTail.next = e;
loTail = e;
++lc;
}
else {
if ((e.prev = hiTail) == null)
hiHead = e;
else
hiTail.next = e;
hiTail = e;
++hc;
}
}
if (loHead != null) {
if (lc <= UNTREEIFY_THRESHOLD)
tab[index] = loHead.untreeify(map);
else {
tab[index] = loHead;
if (hiHead != null) // (else is already treeified)
loHead.treeify(tab);
}
}
if (hiHead != null) {
if (hc <= UNTREEIFY_THRESHOLD)
tab[index + bit] = hiHead.untreeify(map);
else {
tab[index + bit] = hiHead;
if (loHead != null)
hiHead.treeify(tab);
}
}
}
// 左旋操作
static <K,V> TreeNode<K,V> rotateLeft(TreeNode<K,V> root,
TreeNode<K,V> p) {
TreeNode<K,V> r, pp, rl;
if (p != null && (r = p.right) != null) {
if ((rl = p.right = r.left) != null)
rl.parent = p;
if ((pp = r.parent = p.parent) == null)
(root = r).red = false;
else if (pp.left == p)
pp.left = r;
else
pp.right = r;
r.left = p;
p.parent = r;
}
return root;
}
//右旋操作
static <K,V> TreeNode<K,V> rotateRight(TreeNode<K,V> root,
TreeNode<K,V> p) {
TreeNode<K,V> l, pp, lr;
if (p != null && (l = p.left) != null) {
if ((lr = p.left = l.right) != null)
lr.parent = p;
if ((pp = l.parent = p.parent) == null)
(root = l).red = false;
else if (pp.right == p)
pp.right = l;
else
pp.left = l;
l.right = p;
p.parent = l;
}
return root;
}
static <K,V> TreeNode<K,V> balanceInsertion(TreeNode<K,V> root,
TreeNode<K,V> x) {
x.red = true;
for (TreeNode<K,V> xp, xpp, xppl, xppr;;) {
if ((xp = x.parent) == null) {
x.red = false;
return x;
}
else if (!xp.red || (xpp = xp.parent) == null)
return root;
if (xp == (xppl = xpp.left)) {
if ((xppr = xpp.right) != null && xppr.red) {
xppr.red = false;
xp.red = false;
xpp.red = true;
x = xpp;
}
else {
if (x == xp.right) {
root = rotateLeft(root, x = xp);
xpp = (xp = x.parent) == null ? null : xp.parent;
}
if (xp != null) {
xp.red = false;
if (xpp != null) {
xpp.red = true;
root = rotateRight(root, xpp);
}
}
}
}
else {
if (xppl != null && xppl.red) {
xppl.red = false;
xp.red = false;
xpp.red = true;
x = xpp;
}
else {
if (x == xp.left) {
root = rotateRight(root, x = xp);
xpp = (xp = x.parent) == null ? null : xp.parent;
}
if (xp != null) {
xp.red = false;
if (xpp != null) {
xpp.red = true;
root = rotateLeft(root, xpp);
}
}
}
}
}
}
static <K,V> TreeNode<K,V> balanceDeletion(TreeNode<K,V> root,
TreeNode<K,V> x) {
for (TreeNode<K,V> xp, xpl, xpr;;) {
if (x == null || x == root)
return root;
else if ((xp = x.parent) == null) {
x.red = false;
return x;
}
else if (x.red) {
x.red = false;
return root;
}
else if ((xpl = xp.left) == x) {
if ((xpr = xp.right) != null && xpr.red) {
xpr.red = false;
xp.red = true;
root = rotateLeft(root, xp);
xpr = (xp = x.parent) == null ? null : xp.right;
}
if (xpr == null)
x = xp;
else {
TreeNode<K,V> sl = xpr.left, sr = xpr.right;
if ((sr == null || !sr.red) &&
(sl == null || !sl.red)) {
xpr.red = true;
x = xp;
}
else {
if (sr == null || !sr.red) {
if (sl != null)
sl.red = false;
xpr.red = true;
root = rotateRight(root, xpr);
xpr = (xp = x.parent) == null ?
null : xp.right;
}
if (xpr != null) {
xpr.red = (xp == null) ? false : xp.red;
if ((sr = xpr.right) != null)
sr.red = false;
}
if (xp != null) {
xp.red = false;
root = rotateLeft(root, xp);
}
x = root;
}
}
}
else { // symmetric
if (xpl != null && xpl.red) {
xpl.red = false;
xp.red = true;
root = rotateRight(root, xp);
xpl = (xp = x.parent) == null ? null : xp.left;
}
if (xpl == null)
x = xp;
else {
TreeNode<K,V> sl = xpl.left, sr = xpl.right;
if ((sl == null || !sl.red) &&
(sr == null || !sr.red)) {
xpl.red = true;
x = xp;
}
else {
if (sl == null || !sl.red) {
if (sr != null)
sr.red = false;
xpl.red = true;
root = rotateLeft(root, xpl);
xpl = (xp = x.parent) == null ?
null : xp.left;
}
if (xpl != null) {
xpl.red = (xp == null) ? false : xp.red;
if ((sl = xpl.left) != null)
sl.red = false;
}
if (xp != null) {
xp.red = false;
root = rotateRight(root, xp);
}
x = root;
}
}
}
}
}
/**
* Recursive invariant check
*/
static <K,V> boolean checkInvariants(TreeNode<K,V> t) {
TreeNode<K,V> tp = t.parent, tl = t.left, tr = t.right,
tb = t.prev, tn = (TreeNode<K,V>)t.next;
if (tb != null && tb.next != t)
return false;
if (tn != null && tn.prev != t)
return false;
if (tp != null && t != tp.left && t != tp.right)
return false;
if (tl != null && (tl.parent != t || tl.hash > t.hash))
return false;
if (tr != null && (tr.parent != t || tr.hash < t.hash))
return false;
if (t.red && tl != null && tl.red && tr != null && tr.red)
return false;
if (tl != null && !checkInvariants(tl))
return false;
if (tr != null && !checkInvariants(tr))
return false;
return true;
}
