CS 350 : Data Structures AVL Trees
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1 CS 35 : Data Structures AVL Trees David Babcock (courtesy of James Moscola) Department of Physical Sciences York College of Pennsylvania CS 35: Data Structures James Moscola
2 Balanced Search Trees Binary search trees are not guaranteed to be balanced given random insertions and deletions - Inserting a sorted lists of elements into a BST produces the worst case -- O(N) - Performance of an unbalanced tree can degrade as more elements are inserted Balanced search tree operations, such as insert, insure that the a tree always remains balanced - An operation is not complete until it returns the tree to a balanced state Balanced tree Unbalanced tree CS 35: Data Structures 2
3 AVL Trees A type of balanced binary search tree Named for its discoverers -- Adelson, Velskii, and Landis Definition: An AVL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. CS 35: Data Structures 3
4 AVL Trees 2 3 The height of each node is listed in blue An AVL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. CS 35: Data Structures 4
5 AVL Trees 2 3 The height of each node is listed in blue height differs by at most An AVL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. CS 35: Data Structures 5
6 AVL Trees 2 3 The height of each node is listed in blue height differs by at most An AVL tree is a binary search tree with the additional balance property that, for any node in the tree, the height of the left and right subtrees can differ by at most. The height of an empty subtree is. CS 35: Data Structures 6
7 AVL Trees The binary tree after inserting node 2 What are the node heights CS 35: Data Structures 7
8 AVL Trees The binary tree after inserting node 4 2 What are the node heights CS 35: Data Structures 8
9 AVL Trees 4 2 height differs by more than therefore, node 8 is invalid CS 35: Data Structures 9
10 AVL Trees 4 2 height differs by more than therefore, node 8 is unbalanced height differs by more than therefore, node 2 is unbalanced 2 6 CS 35: Data Structures
11 AVL Trees When a node in the tree no longer satisfies the invariant required to be an AVL tree the tree must be rebalanced around that node There are four ways in which an insertion into a tree may cause an imbalance to occur at some node X: () An insertion in the left subtree of the left child of X (2) An insertion in the right subtree of the left child of X (3) An insertion in the left subtree of the right child of X (4) An insertion in the right subtree of the right child of X symmetric CS 35: Data Structures
12 Creating an Imbalance -- Case # () An insertion in the left subtree of the left child of X N2 N2 2 N N C C A B B root of A subtree increases height as does N and N2 A Node N and root of C differ by more than therefore N2 violates AVL rules CS 35: Data Structures 2
13 Fixing the Imbalance -- Case # Perform a single rotation N becomes new root node N2 2 N N2 now right child of N N N2 C A B B C A B subtree now left child of N2 CS 35: Data Structures 3
14 Creating an Imbalance -- Case #4 (symmetric with ) (4) An insertion in the right subtree of the right child of X N2 N N2 N N A A B C root of C subtree increases height as does N and N2 Node N and root of A differ by more than therefore N2 violates AVL rules B C CS 35: Data Structures 4
15 Fixing the Imbalance -- Case #4 (symmetric with ) Perform a single rotation N2 N2 now left child of N N N becomes new root node N N2 A B C A B C B subtree now right child of N2 CS 35: Data Structures 5
16 Creating an Imbalance -- Case #2 (2) An insertion in the right subtree of the left child of X N3 2 N3 N N C C A B A Node N and root of C differ by more than therefore N3 violates AVL rules B root of B subtree increases height as does N and N3 CS 35: Data Structures 6
17 Fixing the Imbalance -- Case #2 We know the node N2 exists because a node was just inserted into the B subtree N3 2 2 N3 N N C N2 C A A B B B2 CS 35: Data Structures 7
18 Fixing the Imbalance -- Case #2 Perform a double rotation N3 2 Step # - Perform a single rotation between N and N2 N3 2 tree is still unbalanced N N2 N2 C N C A B2 B B2 A B CS 35: Data Structures 8
19 Fixing the Imbalance -- Case #2 Perform a double rotation N3 2 Step #2 - Perform a single rotation between N2 and N3 N2 N2 N N3 N C B2 A B B2 C A B CS 35: Data Structures 9
20 Fixing the Imbalance -- Case #2 Another look at the double rotation without the intermediate steps N3 N2 N N N3 N2 C A A B B2 C B B2 CS 35: Data Structures 2
21 Fixing the Imbalance -- Case #3 (symmetric with 2) Another look at the double rotation without the intermediate steps N3 N2 N N3 N A N2 C A B B2 C B B2 CS 35: Data Structures 2
22 AVL Tree Insertion Example CS 35: Data Structures 22
23 AVL Tree Insertion Example Inserting node - First, recursively search for the location which to insert the node - Next, insert the node - Finally, unwind the recursion update node heights along the way. Rebalance tree where necessary To update node s height, check its children CS 35: Data Structures 23
24 AVL Tree Insertion Example node is a leaf node so its height is 2 6 CS 35: Data Structures 24
25 AVL Tree Insertion Example node 2 - add to the height of its highest child CS 35: Data Structures 25
26 AVL Tree Insertion Example 2 node 4 - add to the height of its highest child CS 35: Data Structures 26
27 AVL Tree Insertion Example node 8 violates AVL properties, so rebalance 2 height differs by more than CS 35: Data Structures 27
28 AVL Tree Insertion Example Case #: Must perform single right rotation between node 8 and node CS 35: Data Structures 28
29 AVL Tree Insertion Example After rotation, update height of node 8 Node 4 is balanced, so continue unwinding recursion CS 35: Data Structures 29
30 AVL Tree Insertion Example node 2 - add to the height of its highest child CS 35: Data Structures 3
31 Other Operations: find / remove The find operation is the same as the unbalanced binary search tree The remove operation works similarly to the remove operation from the unbalanced binary search tree with a few modifications - When a node is removed, the heights of its ancestors may need to be updated as the recursion is unwound -- fix imbalances as they are encountered just like with insertion CS 35: Data Structures 3
32 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) CS 35: Data Structures 32
33 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) CS 35: Data Structures 32
34 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) CS 35: Data Structures 32
35 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) CS 35: Data Structures 32
36 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) CS 35: Data Structures 32
37 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) CS 35: Data Structures 32
38 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) remove O(log N) O(log N) CS 35: Data Structures 32
39 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) remove O(log N) O(log N) CS 35: Data Structures 32
40 Analysis of AVL Tree Operations Time complexity of AVL Tree operations worst case average find O(log N) O(log N) insert O(log N) O(log N) remove O(log N) O(log N) CS 35: Data Structures 32
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