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AVL TREES

AVL TREES. COP 3502. Deletion from an AVL Tree. First, do a normal Binary Search Tree Delete: If the node is a leaf, remove it. If it has 1 child, replace with its child If it has 2 children, replace with the largest in its left subtree ( inorder predecessor) and remove that node.

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AVL TREES

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  1. AVL TREES COP 3502

  2. Deletion from an AVL Tree • First, do a normal Binary Search Tree Delete: • If the node is a leaf, remove it. • If it has 1 child, replace with its child • If it has 2 children, replace with the largest in its left subtree (inorder predecessor) and remove that node. • After deletion, retrace the path back up the tree, starting with the parent of the replacement, to the root, adjusting the balance factor as needed.

  3. Deletion from an AVL Tree • In an insert there is at most one node that needs to be rebalanced. • But in a delete there may be multiple nodes to be rebalanced. • Technically only one rebalance that happens at a node, but once that happens it may affect the ancestral nodes.

  4. Deletion Example • Delete 8: 32 16 48 8 24 40 56 28 36 44 52 60 58 62

  5. Deletion Example 32 16 48 24 40 56 28 36 44 52 60 58 62

  6. Deletion Example A 32 32 B 24 48 48 C 16 16 28 40 56 56 36 44 52 60 58 62

  7. Deletion Example 48 32 56 24 52 60 40 58 62 16 16 28 36 44

  8. Choosing A,B,C for Delete Restructuring • One thing that is more complicated about choosing the nodes A, B and C for the AVL Tree delete restructuring is that these nodes are NOT from the ancestral path followed from the origin of the delete. • Clearly, if a delete will cause an imbalance, it will be because the subtree that contains the deleted node has become too short. • Remember that the nodes A, B and C are always on the “longest” path to the bottom of the tree. • This means that when we find an imbalanced node after deleting, the node to the opposite side is guaranteed to be down the longer path. BF=2 30 A 15 45 B 10 35 50 C 32 55

  9. Choosing A,B,C for Delete Restructuring • After labeling the first two nodes of A,B,C and there is still a choice between the right and left: • If one side is longer than the other, choose that side. • If the two sides are equal, go to the same side as the parent is to the grandparent. BF=2 30 B 45 A A 15 45 B 30 C 50 C 10 35 50 15 35 55 32 55 10 32

  10. Choosing A,B,C for Delete Restructuring • The following situation is similar if we delete 50 from the following tree: BF=2 B C 15 30 A B 15 45 10 30 C A 50 10 17 5 17 45 5 20 20

  11. AVL Tree Delete Examples • The most simple example is when a node from a tree with four nodes gets deleted. • Consider deleting 12 from the following tree: C 10 10 B C 8 A 5 12 B 8 A 5 10 C 8 B A 5

  12. AVL Tree Delete Examples • Delete 30 from the following tree: BF=2 C 20 C 20 B 15 A 10 30 25 B 15 30 25 A 10 C 20 5 15 25 A B 10 5 12 25 12 5 12

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