ADVANCED DATA STRUCTURES STUDY NOTES. The left subtree of each node contains values that are smaller than the value in the given node.

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1 UNIT 2 TREE STRUCTURES ADVANCED DATA STRUCTURES STUDY NOTES Binary Search Trees- AVL Trees- Red-Black Trees- B-Trees-Splay Trees. HEAP STRUCTURES: Min/Max heaps- Leftist Heaps- Binomial Heaps- Fibonacci Heaps. Binary Search Trees The left subtree of each node contains values that are smaller than the value in the given node. The right subtree of each node contains values that are greater than the value in the given node. Example: Operations: 1. Search - compare the values and proceed either to the left or to the right. 2. Insert - unsuccessful search - insert the new node at the bottom where the search has stopped.

2 3. Delete - replace the value in the node with the smallest value in the right subtree or the largest value in the left subtree.

3 The idea behind is: all the values to the right are greater than all the values to the left, so the smallest one will be greater too (i.e. after the replacement its left subtree will contain smaller values.) And since we have taken the smallest from the right subtree, after the replacement the right subtree will contain only greater values. Creating a binary search tree: Initially the tree is empty. The first element to be stored is placed in a node that would be the root of the tree. Each next element is placed through the "unsuccessful search" procedure. Retrieving the elements in sorted order - by in-order traversal. Complexity of search in a binary tree: O(logN) The search at worst will proceed to the leaves, down a path in the tree. On average the depth of a binary tree with N elements is log(n). This is not the worst case though. The worst case is when the items to be inserted are sorted, then the tree degenerates to a list: The tree is not balanced. Disadvantages of Binary Search Trees:

4 The shape of the tree depends on the order of insertions, and it can be degenerated. When inserting or searching for an element, the key of each visited node has to be compared with the key of the element to be inserted/found. Keys may be long and the run time may increase much. 1. AVL Trees - Rotating the nodes If thetree is unbalanced, the complexity mayincrease to O(N). The method to keepthe tree balancedis called node rotation. AVL Trees are binary search trees that use node rotation upon inserting a new node. (Adelson-Velskii and Landis) AVL idea: keep the tree in balance condition - the left and the right subtrees of each node should differ by at most one level. It can be proved that if this condition is observed the depth of the tree is O(logN). Rotation of nodes When new nodes are inserted the balance condition might be violated. To restore the balance we rotate the nodes: When we insert a new node, one of the subtrees may become longer by 2 than the other. We shall call this subtree - violated subtree, and the node which has this subtree - violated node. The violated subtree may be left or right subtree of the violated node. In its turn the violated subtree has a root and two subtrees: left -we'll call it LS, and right - RS. The new node has been inserted either in LS or in RS. Two cases are considered: 1. The subtree where the new node is inserted is on the same side as the violated subtree., e.g. the new node has been inserted in LS, and the violated subtree is a left subtree. 2. The subtree where the new node is inserted is on different side, e.g. the new node has been inserted in LS, and the violated subtree is a right subtree. Example: Same side:

5 Different side: Single rotation: to fix the tree in the first case

6 The violated node has to be lowered, and the root of the violated tree - moved up. A - the violated node Tree1 - right B - root of Tree1 Tree2 - its right C - root of the left subtree of B subtree that causes the balance violation subtree, where the new node is inserted Rotate nodes A and B Nodes whose links are going to change: Parent of A, A, B Rearrangement of links: WAS: Parent of A A_right B B_left C BECOMES: Parent of B A_right C B_left A New Tree after rotation

7 Double rotation

8 Tree1 - right subtree (the violated tree) Tree2 - left subtree (the subtree of Tree1 where the new node is inserted) The sides are different, so we need double rotation. What happens if we do a single rotation between node A and node B? Tree2 becomes a right subtree of node A, the new node (with the additional link) is in Tree2, and since node A is lowered, the balance is not restored. The method: Do a single rotation between node C and node B. Node C will become the root of Tree1. As a result Tree2 will change - its depth will be decreased, at the expense of increasing the depth of the right subtree of Tree1. This is now the first case - the violated tree and its subtree with increased depth will have same side - they are both right subtrees. We can apply single rotation between the nodes A and C (the new root of Tree1)

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