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AVL Trees

And the pain you must suffer to learn them. AVL Trees. The AVL Property. For every node the left and right side differ in height by less than two. The left and right side are themselves AVL trees. How Balanced is That?. AVL trees might end up kinda 'sparse'.

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AVL Trees

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  1. And the pain you must suffer to learn them AVL Trees

  2. The AVL Property • For every node the left and right side differ in height by less than two. • The left and right side are themselves AVL trees.

  3. How Balanced is That? • AVL trees might end up kinda 'sparse'. • The worst case height of an AVL tree with n nodes is about 1.44log(n). • Discussion at http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/AVL.html

  4. Rotations • Rotations are to rearrange nodes to maintain AVLness. • They are needed on insert and delete.

  5. Names of Rotations • There are four of them • Single Left (LL) and Single Right (RR) • Double Left (LR) and Double Right (RL) • The names describe which way the node moves. For example, a left rotation moves the node down and left. • Nodes always move down when rotated.

  6. How To Do Single Rotations

  7. A Left Rotation via Pointers temp=p->right ; p->right=temp->left ; temp->left=p ; p=temp ;

  8. Double Rotation • A LR double rotation is to rotate something to the left, and then its former parent to the right. • A RL double rotation is to rotate something to the right, and then its former parent to the left.

  9. Rotation Examples • All the rotations in picture form http://sky.fit.qut.edu.au/~maire/avl/System/AVLTree.html • All the rotations in pictures and text http://www.cs.jcu.edu.au/Subjects/cp2001/1997/foils/balancedTrees/balancedTrees.html • Sample single and double rotationshttp://ironbark.bendigo.latrobe.edu.au/courses/subjects/DataStructures/mal/session200/lecture.html

  10. The AVL Insertion Algorithm • An Intuitive description can be found at http://www.ucfv.bc.ca/cis/watkissc/200009/200009Comp175/notes/avlTrees.html

  11. The AVL Game • Play the game http://www.seanet.com/users/arsen/avltree.html

  12. The Insertion Algorithm • Recursively call down to find the insertion point. • Insert there (it will always be a leaf node). • Call rebalance at the end. • Will rebalance from the bottup up. • Will rebalance only the path from the change to the root.

  13. Insertion Code insert_node (inData) { if (this == NULL) return new Node<T> (inData); if (inData < data) left = left -> insert_node (inData); else right = right -> insert_node (inData); return balance (); }

  14. Notes About the Code • Calls balance on the way up. • Only calls balance on the path between the change and the root. • Call balance too many times for an insert. • In this example ....We never change the parent node (or root).Instead we return what that parent node should be. Neat trick!

  15. The Deletion Algorithm • Find the place to delete. • Swap with the in order successor. • Delete that node • Remember, he might have ONE kid. • Rebalance on the way up from the node you just deleted.

  16. Balancing Let d = difference in height between kids. If (abs(d) < 2) return; If (d < 0) // Heavy on the left if (right->difference > 0) // kid heavy on right right = right->rotate_right(); return rotate_left(); .... // same stuff other side recompute my height return

  17. Computing the Height • Each node's height is just max(left->height, right->height) + 1; • This only changes along the path from insertion/deletion to the root. • The obvious recursive code searches the entire tree ... don't do that!

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