Binary Search Trees

1
Binary Search Trees

• Irena Pevac
• CS501

2
BST

• Binary Search Tree is special type of a binary
  tree where keys come from an ordered set. For
  every node all keys in a left subtree are less
  than or equal to the key in a node and all keys
  in the right subtree are larger than the key in a
  node.

3
BST Properties

• An inorder traversal of a BST produces a sorted
  list of keys.
• BST trees can vary greatly in their degree of
  balance.
• Example that follows shows two BST trees with the
  same set of keys, one is balanced and other is
  not.

4
Two BSTs
5
Basic Operations

• Search
• To search for an item we move through one path in
  the tree and if successful we will find it
  somewhere on that path if not we stop when we
  reach a leaf node.
• Insert
• To insert an item we move through one path in the
  tree and insert new item as a leaf.
• Delete
• Delete a leaf
• Delete a node with one empty subtree
• Delete a node with two nonempty subtrees

6
Delete a Leaf

• Make a reference from the parent of the node to
  be deleted to be null.

7
Leaf Deleted
8
Delete a Node With One Child
9
Node With One Child Deleted
10
Delete a Node With Two Children
11
Find Smallest in Right Subtree
12
Node With Two Children Deleted
13
BST Search

• Element bstSearch(BinTree bst, Key K)
• Element found
• if (bst nil)
• found null
• else
• Element root root(bst)
• If (K root.key)
• found root
• else if (K lt root.key)
• found bstSearch (leftSubtree(bst), K)
• else
• found bstSearch(rightSubtree(bst), K)
• return found

14
Analysis of Binary Search Tree Search

• Basic operation the number of internal nodes of
  the tree that are examined while searching for a
  key. Comparing a key twice is counted as a two
  way comarison.
• For unbalaced BST, cost is ?(n)
• For a balanced BST, cost is ?(lg n)
• We try to make the tree balanced using
  technique of binary tree rotations

15
Rotation

• Rotation involves a group of two connected nodes,
  say p, and c for parent and child.

16
Binary Tree Rotations
17
Left Rotation

• In a left rotation p is to the left of c.
• Node p sinks
• Node c rises
• The edge to the middle subtree moves to the left
  to hook up with p.
• The left principal subtree sinks along with p.
• The right principal subtree rises along with c.
• The middle principal subtree remains on the same
  level.