红黑树和b+树一样，是平衡二叉搜索树。红黑树每个节点都是有颜色的，要么是红色，要么黑色，但树的根是黑色，最底部的叶也是黑色的。还需要注意的是，红黑树任何节点到叶的直接路径包含相同数量的黑色节点。
红黑树如何保持自平衡的特性？
红黑树节点颜色的限制确保从根到叶的最长路径不超过最短路径的两倍。
为什么新插入的节点在红黑树中总是红色的？
这是因为插入红色节点不会违反红黑树的黑色节点数量性质。而且即便是新增红色节点插入到原本的红色节点，解决此问题会比违反黑色节点引起的问题更加容易。
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红黑树Python代码实现
import sys
# 创建节点
class Node():
def __init__(self, item):
self.item = item
self.parent = None
self.left = None
self.right = None
self.color = 1
class RedBlackTree():
def __init__(self):
self.TNULL = Node(0)
self.TNULL.color = 0
self.TNULL.left = None
self.TNULL.right = None
self.root = self.TNULL
# 前序
def pre_order_helper(self, node):
if node != TNULL:
sys.stdout.write(node.item + " ")
self.pre_order_helper(node.left)
self.pre_order_helper(node.right)
# 中序
def in_order_helper(self, node):
if node != TNULL:
self.in_order_helper(node.left)
sys.stdout.write(node.item + " ")
self.in_order_helper(node.right)
# 后根
def post_order_helper(self, node):
if node != TNULL:
self.post_order_helper(node.left)
self.post_order_helper(node.right)
sys.stdout.write(node.item + " ")
# 搜索树
def search_tree_helper(self, node, key):
if node == TNULL or key == node.item:
return node
if key < node.item:
return self.search_tree_helper(node.left, key)
return self.search_tree_helper(node.right, key)
# 删除后平衡树
def delete_fix(self, x):
while x != self.root and x.color == 0:
if x == x.parent.left:
s = x.parent.right
if s.color == 1:
s.color = 0
x.parent.color = 1
self.left_rotate(x.parent)
s = x.parent.right
if s.left.color == 0 and s.right.color == 0:
s.color = 1
x = x.parent
else:
if s.right.color == 0:
s.left.color = 0
s.color = 1
self.right_rotate(s)
s = x.parent.right
s.color = x.parent.color
x.parent.color = 0
s.right.color = 0
self.left_rotate(x.parent)
x = self.root
else:
s = x.parent.left
if s.color == 1:
s.color = 0
x.parent.color = 1
self.right_rotate(x.parent)
s = x.parent.left
if s.right.color == 0 and s.right.color == 0:
s.color = 1
x = x.parent
else:
if s.left.color == 0:
s.right.color = 0
s.color = 1
self.left_rotate(s)
s = x.parent.left
s.color = x.parent.color
x.parent.color = 0
s.left.color = 0
self.right_rotate(x.parent)
x = self.root
x.color = 0
def __rb_transplant(self, u, v):
if u.parent == None:
self.root = v
elif u == u.parent.left:
u.parent.left = v
else:
u.parent.right = v
v.parent = u.parent
# 节点删除
def delete_node_helper(self, node, key):
z = self.TNULL
while node != self.TNULL:
if node.item == key:
z = node
if node.item <= key:
node = node.right
else:
node = node.left
if z == self.TNULL:
print("Cannot find key in the tree")
return
y = z
y_original_color = y.color
if z.left == self.TNULL:
x = z.right
self.__rb_transplant(z, z.right)
elif (z.right == self.TNULL):
x = z.left
self.__rb_transplant(z, z.left)
else:
y = self.minimum(z.right)
y_original_color = y.color
x = y.right
if y.parent == z:
x.parent = y
else:
self.__rb_transplant(y, y.right)
y.right = z.right
y.right.parent = y
self.__rb_transplant(z, y)
y.left = z.left
y.left.parent = y
y.color = z.color
if y_original_color == 0:
self.delete_fix(x)
# 插入后平衡树
def fix_insert(self, k):
while k.parent.color == 1:
if k.parent == k.parent.parent.right:
u = k.parent.parent.left
if u.color == 1:
u.color = 0
k.parent.color = 0
k.parent.parent.color = 1
k = k.parent.parent
else:
if k == k.parent.left:
k = k.parent
self.right_rotate(k)
k.parent.color = 0
k.parent.parent.color = 1
self.left_rotate(k.parent.parent)
else:
u = k.parent.parent.right
if u.color == 1:
u.color = 0
k.parent.color = 0
k.parent.parent.color = 1
k = k.parent.parent
else:
if k == k.parent.right:
k = k.parent
self.left_rotate(k)
k.parent.color = 0
k.parent.parent.color = 1
self.right_rotate(k.parent.parent)
if k == self.root:
break
self.root.color = 0
# Printing the tree
def __print_helper(self, node, indent, last):
if node != self.TNULL:
sys.stdout.write(indent)
if last:
sys.stdout.write("R----")
indent += " "
else:
sys.stdout.write("L----")
indent += "| "
s_color = "RED" if node.color == 1 else "BLACK"
print(str(node.item) + "(" + s_color + ")")
self.__print_helper(node.left, indent, False)
self.__print_helper(node.right, indent, True)
def preorder(self):
self.pre_order_helper(self.root)
def inorder(self):
self.in_order_helper(self.root)
def postorder(self):
self.post_order_helper(self.root)
def searchTree(self, k):
return self.search_tree_helper(self.root, k)
def minimum(self, node):
while node.left != self.TNULL:
node = node.left
return node
def maximum(self, node):
while node.right != self.TNULL:
node = node.right
return node
def successor(self, x):
if x.right != self.TNULL:
return self.minimum(x.right)
y = x.parent
while y != self.TNULL and x == y.right:
x = y
y = y.parent
return y
def predecessor(self, x):
if (x.left != self.TNULL):
return self.maximum(x.left)
y = x.parent
while y != self.TNULL and x == y.left:
x = y
y = y.parent
return y
def left_rotate(self, x):
y = x.right
x.right = y.left
if y.left != self.TNULL:
y.left.parent = x
y.parent = x.parent
if x.parent == None:
self.root = y
elif x == x.parent.left:
x.parent.left = y
else:
x.parent.right = y
y.left = x
x.parent = y
def right_rotate(self, x):
y = x.left
x.left = y.right
if y.right != self.TNULL:
y.right.parent = x
y.parent = x.parent
if x.parent == None:
self.root = y
elif x == x.parent.right:
x.parent.right = y
else:
x.parent.left = y
y.right = x
x.parent = y
def insert(self, key):
node = Node(key)
node.parent = None
node.item = key
node.left = self.TNULL
node.right = self.TNULL
node.color = 1
y = None
x = self.root
while x != self.TNULL:
y = x
if node.item < x.item:
x = x.left
else:
x = x.right
node.parent = y
if y == None:
self.root = node
elif node.item < y.item:
y.left = node
else:
y.right = node
if node.parent == None:
node.color = 0
return
if node.parent.parent == None:
return
self.fix_insert(node)
def get_root(self):
return self.root
def delete_node(self, item):
self.delete_node_helper(self.root, item)
def print_tree(self):
self.__print_helper(self.root, "", True)
if __name__ == "__main__":
bst = RedBlackTree()
bst.insert(55)
bst.insert(40)
bst.insert(65)
bst.insert(60)
bst.insert(75)
bst.insert(57)
bst.print_tree()
print("\nAfter deleting an element")
bst.delete_node(40)
bst.print_tree() 
