1. AVL Trees Neil Ghani University of Strathclyde

2. General Trees Recall a tree is * A leaf storing an integer * A node storing a left subtree, an integer and a right subtree. We define algorithms by saying what they do to leaves and what they do to nodes

3. Example – searching a tree For example, we can search a tree for data find z (leaf x) = (x == z) find z (node l x r) = find z l or (z == x) or find z r This is clearly a O(n) algorithm. Can we do better and find a logarithmic algorithm

4. Binary Search Trees With ordered lists we can search in logarithmic time. An ordered tree is called a binary search tree (BST) A tree of the form leaf x is a BST A tree of the form (node l x r) is a BST if * l and r are BSTs * max l <= x * min r >= x

5. Searching a BST Searching a BST can use the fact that a BST is ordered findBST z (leaf x) = (x == z) findBST z (node l x r) = if (z == x) then true else if (z <= x) then findBST z l else findBST z r

6. Complexity of findBST There is one recursive call * If the left and right subtrees are about the same size, T(n) = 2 + T(n/2) * If the right subtree is a leaf and we search the left subtree, T(n) = 2 + T(n-2) Worst case complexity is linear

7. AVL Trees AVL trees ensure that the left and right subtrees are about equal size and hence allow logarithmic searching. Before talking about AVL trees, note that there are no trees, or BSTs, with 2 elements. To rectify this, we say that there is another type of tree called the empty tree, written E

8. Some trees with empty Node E 5 (leaf 4) represents the tree 5 / \ E 4 Node (leaf 4) 5 E represents the tree 5 / \ 4 E

9. More trees with empty Qn: What is the representation of 6 / \ 5 E / \ 2 E Ans: node (node (leaf 2) 5 E) 6 E

10. Definition of AVL trees A tree of the form (leaf x) is an AVL tree A tee of the form E is an AVL tree A tree of the form node l x r is an AVL tree if * node l x r is a BST * l and r are AVL trees * |height l – height r| <= 1 AVL trees are also called balanced trees

11. Definition of height Height computes the height of a tree height E = 0 height (leaf x) = 1 height (node l x r) = 1 + max (height l, height r)

12. Creating AVL trees There are no free lunches * Searching AVL trees is logarithmic * Creating AVL trees is harder … Question: How to add a piece of data to an AVL tree in such a way to get an AVL tree * add data to make a BST * balance the result

13. Adding data to a BST Here is the algorithm. Do you understand it? addBST x E = leaf x addBST x (leaf z) = if x <= z then node (leaf x) z E else node E z (leaf x) addBST x (node l z r) = if x <= z then node (addBST x l) z r else node l z (addBST x r)

14. Creating an AVL tree addBST, when applied to an AVL tree, creates a BST but not an AVL tree. 4 problem cases … here are 2 with the other 2 being reflections. 5 5 / \ / \ 3 D 3 D / \ / \ A 4 2 C / \ / \ B C A B

15. Rotations The solution is to use rotations * first case is called left-right. Rotate the left subtree to obtain the second case * second case is called the left-left case. Rotate the whole tree to get an AVL tree Why is the result a BST?

16. Red-black Trees A different notion of balancing * Not as balanced as AVL trees, but * Easier insertion Key property: In a red/black tree t longpath t <= 2 * shortpath t where longpath is the longest path from the root to a leaf and shortpath is the shortest path,

17. Definition of Red/Black trees A BST is a red/black tree if * All nodes are coloured red or black * All leaves are black * Children of red nodes are black * Given a node in the tree, all paths from the node to a leaf contain the same number of black nodes.