AVL Trees

Neil GhaniUniversity of Strathclyde

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

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

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

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

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

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

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

More trees with empty

• Qn: What is the representation of 6 / \ 5 E / \ 2 E • Ans: node (node (leaf 2) 5 E) 6 E

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

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)

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

Adding data to a BST

• Here is the algorithm. Do you understand it?addBST x E = leaf xaddBST 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)

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

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?

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,

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.