CIS265/ Trees Red-Black Trees. Some of the following material is from:

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1 CIS265/ Trees Red-Black Trees Some of the following material is from: Data Structures for Java William H. Ford William R. Topp ISBN Chapter 27 Balanced Search Trees Bret Ford 2005, Prentice Hall CIS265/506: Red-Black Trees 2 CIS265/506: Red-Black Trees 1

2 2-3-4 Trees Trees are a slightly less efficient than other ordered tree organizations but they are easier to code and understand. As in most of the self-balanced trees, they facilitate searching, insertions and deletions in the order of O(log N) operations regardless of how data values are entered. CIS265/506: Red-Black Trees Tree Concepts Each node has 2-to-4 outgoing links. This means that there are 0-3 data items in a node. The number of links is referred to as the order of the tree 2-Node: [ptr, A, ptr] 3-Node: [ptr, A, ptr, B, ptr] 4-node: [ptr, A, ptr, B, ptr, C, ptr] ptr1 A Ptr2 B ptr3 C ptr4 < A < B < C > C CIS265/506: Red-Black Trees 4 CIS265/506: Red-Black Trees 2

3 2-3-4 Tree Concepts Data items in the node are stored in sorted order. The links may have to move depending on insertions or deletions to the data in the node A < B < C ptr1 A Ptr2 B ptr3 C ptr4 CIS265/506: Red-Black Trees Tree Concepts Outgoing Links refer/point to children that are between the data items. End links just compare to the nearest data item (lesser or greater). There will always be one more link than the number of data items ptr1 10 Ptr2 20 ptr3 30 ptr4 ptr1 12 Ptr2 18 ptr3 19 ptr4 ptr1 5 Ptr2 7 ptr3 8 ptr4 6 CIS265/506: Red-Black Trees 3

4 Searching a Tree To find a given value called item, start at the root and compare item with the values in the existing node. If no match occurs, move to the appropriate sub-tree. Repeat the process until you find a match or encounter an empty sub-tree. CIS265/506: Red-Black Trees 7 Searching a Tree Search for value: 6 CIS265/506: Red-Black Trees 8 CIS265/506: Red-Black Trees 4

5 2-3-4 Tree Concepts Inserting into a Tree can be fairly easy or hard, depending on the condition of the nodes on the way to this node If all the nodes on the path are not full, we just need to traverse the tree and insert the data at the leaf level If some nodes on the way are full, we split those nodes and continue if the leaf is full then we split that and move the middle value up CIS265/506: Red-Black Trees Trees In a tree, a 2-node has two children and one value, a 3-node has 3 children and 2 values, and a 4-node has 4 children and 3 values. CIS265/506: Red-Black Trees 10 CIS265/506: Red-Black Trees 5

6 Inserting into a Tree New data items are always inserted in leaves at the bottom of the tree. To insert a new item, move down the tree, splitting any 4-node encountered in the insertion path. Split a node by moving its middle element up one level and creating two new 2-nodes descendants (this includes splitting the root if it is a 4-node see next figure). CIS265/506: Red-Black Trees 11 Inserting into a Tree (continued) Splitting a 4-node. CIS265/506: Red-Black Trees 12 CIS265/506: Red-Black Trees 6

7 Inserting into a Tree (continued) Insert into a tree the following data values: 2, 15, 12, 4, 8, 10, 25, 35, 55, 11 CIS265/506: Red-Black Trees 13 Building a Tree Insert: 2, 15, 12 Insert: 4 Insert: 8, 10 CIS265/506: Red-Black Trees 14 CIS265/506: Red-Black Trees 7

8 Building a Tree (continued) Insert: 25, 35 Insert: 55 CIS265/506: Red-Black Trees 15 Building a Tree (continued) Insert: 55 CIS265/506: Red-Black Trees 16 CIS265/506: Red-Black Trees 8

9 Building a Tree (concluded) Insert: Split 4-node (4, 12, 25) Insert 11 CIS265/506: Red-Black Trees 17 Efficiency of Trees In a tree with n elements, the maximum number of nodes visited during the search for an element is log 2 (n) + 1. Inserting an element into a tree with n elements requires splitting no more than log 2 n nodes and normally requires far fewer splits. CIS265/506: Red-Black Trees 18 CIS265/506: Red-Black Trees 9

10 Red-Black Trees A red-black tree is a binary search tree in which each node has the color attribute BLACK or RED. It was designed as a representation of a tree, using different color combinations to describe the 3-nodes and 4-nodes. It is a type of tree that maintains balance via a set of four rules and associated operations to enforce those rules. intelligent work is done as nodes are inserted as well as when they are deleted CIS265/506: Red-Black Trees 19 The Four Rules Every node in the tree is colored red or black The root is always colored black If a node is red its children are always black Every path to all leaves (filled or waiting to be filled) must go through the same number of black nodes CIS265/506: Red-Black Trees 20 CIS265/506: Red-Black Trees 10

11 Red-Black Trees CIS265/506: Red-Black Trees 21 Representing Tree Nodes A 2-node is always black. A 4-node has the middle value as a black parent and the other values as red children. CIS265/506: Red-Black Trees 22 CIS265/506: Red-Black Trees 11

12 Representing Tree Nodes (concluded) Represent a 3-node with a BLACK parent and a smaller RED left child or with a BLACK parent and a larger RED right child. 3-node (A, B) in a Tree A B (a) Red-black tree representation A is a black parent; B is a red right child A (b) Red-black tree representation B is a black parent; A is a red left child B S T U S T B U S A T U CIS265/506: Red-Black Trees 23 Representing a Tree as a Red-Black Tree CIS265/506: Red-Black Trees 24 CIS265/506: Red-Black Trees 12

13 Properties of a Red-Black Tree These properties follow from the representation of a tree as a red-black tree. Root of red-black tree is always BLACK. A RED parent never has a RED child. Thus in a red-black tree there are never two successive RED nodes. Every path from the root to an empty subtree contains the same number of BLACK nodes. The number, called the black height, defines balance in a red-black tree. CIS265/506: Red-Black Trees 25 Inserting a Node in a Red-Black Tree (continued) Enter a new element into the tree as a RED leaf node. Inserting a RED node at the bottom of a tree may result in two successive RED nodes. When this occurs, use a rotation and recoloring to reorder the tree. Maintain the root as a BLACK node. CIS265/506: Red-Black Trees 26 CIS265/506: Red-Black Trees 13

14 Building a Red-Black Tree CIS265/506: Red-Black Trees 27 Building a Red-Black Tree (continued) CIS265/506: Red-Black Trees 28 CIS265/506: Red-Black Trees 14

15 Building a Red-Black Tree (concluded) Insert 30 CIS265/506: Red-Black Trees 29 Red-Black Tree Search Running Time The worst-case running time to search a redblack tree or insert an item is O(log 2 n). The maximum length of a path in a red-black tree with black height B is 2*B-1. CIS265/506: Red-Black Trees 30 CIS265/506: Red-Black Trees 15

16 Note There is no code in your text for this structure. CIS265/506: Red-Black Trees 31 CIS265/506: Red-Black Trees 16

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