CSE 3358 Note Set 12

1. CSE 3358 Note Set 12 Data Structures and Algorithms

2. Exam 1 Next Wednesday 9/24/08

3. Deletion by Copying 13 25 20 31 23 39 21 24

4. template<class T> void BST<T>::deleteByCopying(BSTNode<T>*& node) { BSTNode<T> *previous, *tmp = node; if (node->right == 0) node = node->left; else if (node->left == 0) node = node->right; else { tmp = node->left; previous = node; while (tmp->right != 0) { previous = tmp; tmp = tmp->right; } node->key = tmp->key; if (previous == node) previous->left = tmp->left; else previous->right = tmp->left; } delete tmp; }

5. Some Different Cases //// see prev slides else { tmp = node->left; previous = node; while (tmp->right != 0) { previous = tmp; tmp = tmp->right; } node->key = tmp->key; if (previous == node) previous->left = tmp->left; else previous->right = tmp->left; } 13 Delete 25 25 20 31 16

6. Some Different Cases //// see prev slides else { tmp = node->left; previous = node; while (tmp->right != 0) { previous = tmp; tmp = tmp->right; } node->key = tmp->key; if (previous == node) previous->left = tmp->left; else previous->right = tmp->left; } 13 Delete 25 25 20 31 23 22

7. Balancing Act • Perfectly Balanced Tree: All leaves are to be found on one level or two levels • For perfectly balanced tree, height = ceil(lg(n)) • Example: 10000 nodes height = ceil(lg(10000)) = ceil(13.289) = 14 • This means, for a perfectly balanced tree, at most 14 nodes have to be checked to locate a particular element

8. How to Balance • Depends on when tree is formed in relation to when searches are performed • If all data is present before tree is formed, • Store in array • Sort using efficient sorting algo • Used binary search type algo to create tree

9. Example • Initial Data Set: • 5 1 9 8 7 0 2 3 4 6 • Sorted: • 0 1 2 3 4 5 6 7 8 9 4 Middle element becomes root Then recursively process right subtree then left subtree

10. Example • Sorted: • 0 1 2 3 4 5 6 7 8 9 4 1

11. Binary Search Balance template <class T> void BST<T>::balance (T data[], int first, int last) { if(first <= last) { int middle = (first + last)/2; insert(data[middle]); balance(data, first, middle – 1); balance(data, middle + 1, last); } }

12. Discussion • All data must be present to create the tree • Subsequent inserts can lead to unbalancing tree • Best case sorting of array is n*lg(n) • Can produce array/list from tree w/ inorder traversal • Space needed? • Then recreate the tree w/ balance() method

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