Building a Red-Black Binary Tree in Python
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A red-black tree is a kind of self-balancing binary search tree. Each node stores an extra bit, which we will call the color, red or black. The color ensures that the tree remains approximately balanced during insertions and deletions. When the tree is modified, the new tree is rearranged and repainted to restore the coloring properties that constrain how unbalanced the tree can become in the worst case.
The purpose of a red-black tree is to stay balanced which ensures that its common operations, like lookup and delete, never degrade to worse than O(n*log(n)).
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What is a balanced binary tree?
Since the reason colors are added to a binary tree is to ensure that it remains balanced, we need to understand how and why a binary tree is balanced. To put it simply, a balanced tree’s branches differ in height by no more than 1.
The following tree is balanced because between its two branches one has a height of 2, and the other 3, meaning they differ by no more than 1.
A / \ B C / DThe next tree is unbalanced because it’s branches differ in height by more than 1. C‘s right side has a height of 2 while its left side has a height of 4).
A / \ B C / / D E / GWhy do we want balanced trees?
Balanced binary search trees ensure speed. The speed of an operation in a binary tree depends on the height of the tree. If the tree is balanced, then the height is only the log of the number of nodes, which means the tree will work as fast as possible. However, if the tree is unbalanced, for example with one really long branch, then the height because the total number of nodes rather than the log.
A / B / C / DProperties of a red-black tree
In addition to all the properties of a Binary Search Tree, a red-black tree must have the following:
- Each node is either red or black
- The root is black. This rule is sometimes omitted. Since the root can always be changed from red to black, but not necessarily vice versa, this rule has little effect on analysis.
- All
nilleaf nodes are black. - If a node is red, then both its children are black.
- All paths from a single node go through the same number of black nodes in order to reach any of its descendant
nilnodes.
Implementing a Red-Black Tree in Python
Step 1 – RBNode Class
Our implementation will use a Tree class and a Node class. The Node will be fairly simple, it’s just a constructor.
class RBNode: def __init__(self, val): self.red = False self.parent = None self.val = val self.left = None self.right = NoneCode language: Python (python)
Step 2 – RBTree Class
Next let’s create a tree class with a constructor.
class RBTree: def __init__(self): self.nil = RBNode(0) self.nil.red = False self.nil.left = None self.nil.right = None self.root = self.nilCode language: Python (python)
Step 3 – Insert method
def insert(self, val): # Ordinary Binary Search Insertion new_node = RBNode(val) new_node.parent = None new_node.left = self.nil new_node.right = self.nil new_node.red = True # new node must be red parent = None current = self.root while current != self.nil: parent = current if new_node.val < current.val: current = current.left elif new_node.val > current.val: current = current.right else: return # Set the parent and insert the new node new_node.parent = parent if parent == None: self.root = new_node elif new_node.val < parent.val: parent.left = new_node else: parent.right = new_node # Fix the tree self.fix_insert(new_node)Code language: Python (python)
The insert method will look a lot like a traditional binary tree insert method. The biggest difference is that after doing an insert, we’ll call a special fix_insert method. For now just call it, we’ll implement it in just a moment.
Step 4 – Rotate left
We’ll need some rotation methods in our “fix” step that’s coming up. Let’s code those now.
# rotate left at node x def rotate_left(self, x): y = x.right x.right = y.left if y.left != self.nil: 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 = yCode language: Python (python)
Step 5 – Rotate right
# rotate right at node x def rotate_right(self, x): y = x.left x.left = y.right if y.right != self.nil: 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 = yCode language: Python (python)
Step 6 – Fix insert
The real bread and butter is in this step, it’s what makes a red-black tree balanced.
def fix_insert(self, new_node): while new_node != self.root and new_node.parent.red: if new_node.parent == new_node.parent.parent.right: u = new_node.parent.parent.left # uncle if u.red: u.red = False new_node.parent.red = False new_node.parent.parent.red = True new_node = new_node.parent.parent else: if new_node == new_node.parent.left: new_node = new_node.parent self.rotate_right(new_node) new_node.parent.red = False new_node.parent.parent.red = True self.rotate_left(new_node.parent.parent) else: u = new_node.parent.parent.right # uncle if u.red: u.red = False new_node.parent.red = False new_node.parent.parent.red = True new_node = new_node.parent.parent else: if new_node == new_node.parent.right: new_node = new_node.parent self.rotate_left(new_node) new_node.parent.red = False new_node.parent.parent.red = True self.rotate_right(new_node.parent.parent) self.root.red = FalseCode language: Python (python)
Full Example of Red-Black Tree in Code
import random class RBNode: def __init__(self, val): self.red = False self.parent = None self.val = val self.left = None self.right = None class RBTree: def __init__(self): self.nil = RBNode(0) self.nil.red = False self.nil.left = None self.nil.right = None self.root = self.nil def insert(self, val): # Ordinary Binary Search Insertion new_node = RBNode(val) new_node.parent = None new_node.left = self.nil new_node.right = self.nil new_node.red = True # new node must be red parent = None current = self.root while current != self.nil: parent = current if new_node.val < current.val: current = current.left elif new_node.val > current.val: current = current.right else: return # Set the parent and insert the new node new_node.parent = parent if parent == None: self.root = new_node elif new_node.val < parent.val: parent.left = new_node else: parent.right = new_node # Fix the tree self.fix_insert(new_node) def fix_insert(self, new_node): while new_node != self.root and new_node.parent.red: if new_node.parent == new_node.parent.parent.right: u = new_node.parent.parent.left # uncle if u.red: u.red = False new_node.parent.red = False new_node.parent.parent.red = True new_node = new_node.parent.parent else: if new_node == new_node.parent.left: new_node = new_node.parent self.rotate_right(new_node) new_node.parent.red = False new_node.parent.parent.red = True self.rotate_left(new_node.parent.parent) else: u = new_node.parent.parent.right # uncle if u.red: u.red = False new_node.parent.red = False new_node.parent.parent.red = True new_node = new_node.parent.parent else: if new_node == new_node.parent.right: new_node = new_node.parent self.rotate_left(new_node) new_node.parent.red = False new_node.parent.parent.red = True self.rotate_right(new_node.parent.parent) self.root.red = False def exists(self, val): curr = self.root while curr != self.nil and val != curr.val: if val < curr.val: curr = curr.left else: curr = curr.right return curr # rotate left at node x def rotate_left(self, x): y = x.right x.right = y.left if y.left != self.nil: 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 # rotate right at node x def rotate_right(self, x): y = x.left x.left = y.right if y.right != self.nil: 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 __repr__(self): lines = [] print_tree(self.root, lines) return '\n'.join(lines) def print_tree(node, lines, level=0): if node.val != 0: print_tree(node.left, lines, level + 1) lines.append('-' * 4 * level + '> ' + str(node.val) + ' ' + ('r' if node.red else 'b')) print_tree(node.right, lines, level + 1) def get_nums(num): random.seed(1) nums = [] for _ in range(num): nums.append(random.randint(1, num-1)) return nums def main(): tree = RBTree() for x in range(1, 51): tree.insert(x) print(tree) main()Code language: Python (python)
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