Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
MCS 360 Lecture 36 Introduction to Data Structures Jan Verschelde, 20 April 2020
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 1 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 2 / 36 Binary Search Trees
Binary search trees store ordered data. Rules to insert x at node N: if N is empty, then put x in N if x < N, insert x to the left of N if x ≥ N, insert x to the right of N
Balanced tree of n elements has depth is log2(n) ⇒ retrieval is O(log2(n)), almost constant. With rotation we make a tree balanced. Alternative to AVL tree: nodes with > 2 children. Every node in a binary tree has at most 2 children. A 2-3-4 tree has nodes with 2, 3, and 4 children. A red-black tree is a binary tree equivalent to a 2-3-4 tree.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 3 / 36 a red-black tree red nodes have hollow rings 11 @ 2 14 @ 1 7 @ 5 8
Four invariants: 1 A node is either red or black. 2 The root is always black. 3 A red node has always black children. 4 #black nodes in path from root to any leaf is the same.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 4 / 36 verification of all four invariants 11 is black 2 is red 1 is black 7 is black 5 is red 8 is red 14 is black The root is black. Node 2 is red and has black left child. Node 2 is red and has black right child. All red nodes have black children. The first path ends at 1 is black. The number of black nodes is 2. Reached end of path at 5 is red. The number of black nodes is 2. Reached end of path at 8 is red. The number of black nodes is 2. Reached end of path at 14 is black. The number of black nodes is 2. The tree respects all invariants.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 5 / 36 a node in a red-black tree template
Node(const Item_Type& item, bool redcolor = true, Node* parent_ptr = NULL, Node* left_ptr = NULL, Node* right_ptr = NULL) : data(item), isred(redcolor), parent(parent_ptr), left(left_ptr), right(right_ptr) {} // Makes a node with as data field the value of item, // the default color is red, and all pointers are set // to NULL by default.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 6 / 36 a class to define a red-black tree
The root of the tree is a pointer to a node, declared as private, as a hidden data attribute.
The bool isred in Node respects the first invariant: all nodes are either red (true) or black (false).
The second invariant: red nodes have black children is verified by a recursive tree traversal.
The #black nodes in any path from root to leaf is the same is verified via backtracking.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 7 / 36 performance
#black nodes in path from root to any leaf is the same ⇒ equivalent 2-3-4 tree is balanced. Performance of a red-black tree:
upper limit of depth of red-black tree: 2 log2(n) + 2 for a search of tree with n elements.
average cost of search is empirically: 1.002 log2(n).
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 8 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 9 / 36 relation with 2-3-4 trees: nodes with 2 and 4 children
No red children: a 2-node
x x @ @ < x≥ x < x≥ x Two red children: a 4-node y x, y, z @ ¨ H x z ¨¨ ¡¡ A HH < x < y ≥ y≥ z ¡ A ¡ A ≥ x < z ¡A ¡A <x ≥x <z ≥z < y ≥ y
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 10 / 36 relation with 2-3-4 trees: nodes with 3 children
One red child: a 3-node y x, y @ ¨ H x ≥ y ¨¨ HH < x ≥ x ≥ y ¡ A < y ¡A <x ≥x < y
One red child: a 3-node x x, y @ ¨ H < xy ¨¨ HH < x ≥ x ≥ y ¡ A < y ¡A <y ≥y ≥ x
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 11 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 12 / 36 Changing Colors
Case 0: The color of a new leaf is always red. If parent of new leaf is black, then done. Case 1: We just change colors, e.g.:
2 2 2 @ @ @ 1 7 1 7 1 7
5 5 5
1 If sibling of parent also red, change color of parent and its sibling to black, and change color of the grandparent. 2 Ensure color of the root is black.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 13 / 36 Tree Rotation
What if parent does not have a red sibling? Case 2: One rotation suffices, e.g.:
2 2 5 @ @ @ 5 5 2 7 @ @ 7 7
1 Change the color of the parent to black, change the color of the grandparent to red. 2 Rotate to restore the 4th invariant.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 14 / 36 Double Rotation
One rotation works for right-right or left-left tree. Case 3: Tree is right-left (or left-right), e.g.:
2 2 2 5 A A A @ 7 5 5 2 7 ¡ A A 5 7 7
1 Rotate right-left tree to right-right tree. 2 Change color of parent and grandparent. 3 Rotate to restore the 4th invariant.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 15 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 16 / 36 inserting into a red-black tree
During insertion, many cases may occur: 1 Locate the new leaf and color it red. 2 If in case 0, then done. 3 Apply cases 1, 2, and 3, restoring invariants.
1 case 1: parent has red sibling ⇒ change colors
2 case 2: left-left or right-right tree ⇒ change colors and rotate
3 case 3: right-left or left-right tree ⇒ rotate, change colors, rotate
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 17 / 36 inserting 4
11 11 @ @ 2 14 2 14 @ @ 1 7 1 7 @ @ 5 8 5 8
4 4
After changing colors of 5 and 8 to black, and changing 7 to red, but then parent of 7 is red too...
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 18 / 36 rotating around 2
11 11 @ @ 2 14 7 14 @ @ 1 7 2 8 @ @ 5 8 1 5
4 4
Case 3: rotate left-right tree (11-2-7) to left-left tree is the first step...
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 19 / 36 changing colors
11 11 @ @ 7 14 7 14 @ @ 2 8 2 8 @ @ 1 5 1 5
4 4
The second step in Case 3 is to change the color of 7 to black and the color of 11 to red.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 20 / 36 rotate around 11
11 @ 7 14 7 @ @ 2 8 2 11 @ ¡A ¡A 1 5 1 5 8 14 ¡ 4 4
In the last step of Case 3, we restored the balance by a right rotation around 11.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 21 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 22 / 36 inserting a sequence of numbers
Insert 48, 84, 43, 91, 38, 79, 63, 59, 49, 98 into a red-black tree. After inserting 48, 84, 43:
48 48 @ @ 43 84 43 84 A 91
After inserting 91 to the binary search tree, we verify the four invariants of a red-black tree.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 23 / 36 a red node has always black children
After inserting 91, we have to change colors:
48 48 @ @ 43 84 43 84 A A 91 91
The numbers left to insert are 38, 79, 63, 59, 49, 98.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 24 / 36 inserting more numbers
Inserting 38, 79, 63:
48 48 @ @ 43 84 43 84 A ¡ ¡A 91 38 79 91 ¡ 63
Are all four invariants of a red-black tree satisfied?
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 25 / 36 changing colors
A red node must always have black children.
48 48 @ @ 43 84 43 84 ¡ ¡A ¡ ¡A 38 79 91 38 79 91 ¡ ¡ 63 63
The numbers left to insert are 59, 49, 98.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 26 / 36 rotate a left-left tree
After interting 59, we have a left-left tree. Coloring 59 black will violate the fourth invariant, so we rotate.
48 48 @ @ 43 84 43 84 ¡ ¡A ¡ ¡A 38 79 91 38 63 91 ¡ ¡A 63 59 79 ¡ 59
The numbers left to insert are 49 and 98.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 27 / 36 inserting 49
After inserting 49, we first change colors:
48 48 @ @ 43 84 43 84 ¡ ¡A ¡ ¡A 38 63 91 38 63 91 ¡A ¡A 59 79 59 79 ¡ ¡ 49 49
Not all invariants of the red-black tree are satisfied.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 28 / 36 rotating once ...
We see that the right tree is left heavy.
48 @ 43 84 48 ¡ ¡A @ 38 63 91 43 63 ¡A ¡ @ 59 79 38 59 84 ¡ ¡ ¡A 49 49 79 91
After rotation, the tree is right heavy.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 29 / 36 rotating again
The new root will become the root of the right tree.
48 63 @ @ 43 63 48 84 ¡ @ ¡A ¡A 38 59 84 43 59 79 91 ¡ ¡A ¡ ¡ 49 79 91 38 49
The left of 63 turned to the right of 48. One number left to insert: 98.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 30 / 36 inserting 98
63 63 @ @ 48 84 48 84 ¡A ¡A ¡A ¡A 43 59 79 91 43 59 79 91 ¡ ¡ ¡ ¡ A 38 49 38 49 98
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 31 / 36 Red-Black Trees
1 Red-Black Trees balancing binary search trees relation with 2-3-4 trees
2 Insertion into a Red-Black Tree algorithm for insertion an elaborate example of an insert inserting a sequence of numbers
3 Recursive Insert Function pseudo code
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 32 / 36 Recursive Insert Function
We extend the node type of a binary search tree: 1 a node contains data (often an integer key); 2 in addition, every node has a color: red or black; 3 two pointers to nodes lead to left and right children.
Prototype of a recursive insert function: bool insert ( Tree &root, Item_Type item );
// returns false if item belonged to root
// returns true if item was added to the root
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 33 / 36 pseudo code
bool insert ( Tree &root, Item_Type item ) if(root == NULL) root = new black node; return true; else if(item == root->data) return false; else if (item < root->data) if(left == NULL) left = new red node; return true; else if((left == red) && (right == red)) change left and right to black and color the local root red;
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 34 / 36 pseudo code continued
if(insert(left,item)) if(left grandchild == red) change left to black and color the local root red; rotate the local root right; else if(right grandchild == red) rotate the left child left; change left to black and color the local root to red; rotate the local root right; else // item > root->data: exercise if(local root is root of the tree) color the root black.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 35 / 36 Summary + Exercises
More on chapter 11 on balancing binary search trees. Exercises: 1 Take the largest example of a red-black tree on these slides and draw the equivalent 2-3-4 tree. 2 Write the pseudo code for the case when the item is inserted to the right child. 3 Instead of inserting 4 in the elaborate example, insert 9 and illustrate all stages in the insertion.
Introduction to Data Structures (MCS 360) Red-Black Trees L-36 20 April 2020 36 / 36