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Lecture COMP 07 Binary Search Trees
Binary Search Trees
Properties:
1. Nodes "value" is greater than all values in its left subtree. 2. Nodes "value" is less than ( ) all values in its right subtree 3. Both the left and right subtrees are also Binary Search Trees
or equal to
Nodes, Records and Search Keys... Oh, my!
Usually used to store a "record" of information. An example record might contain all or more of the following fields:
Name Address Birth date Social Security Number Height, weight, eye color, ethnicity Bank pin number
One field is designated as the "Search Key" and it is this field that determines the ordering and relationship of the nodes in the Binary Search Tree. (May involve more than one field to guarantee unique identification.)
Operations
Insert(newRecord)
The newRecord is inserted in the tree.
For Basic Trees this has the affect of creating and attaching a new leaf somewhere in the tree without disturbing the ordering of the rest of the tree. Implementations that guarantee balanced properties may rotate portions of the tree around during insertion to maintain balance.
Delete(someKey)
Removes the node(s) of the tree that match the search key "someKey".
Retreive(searchKey)
Returns the item in a BST that matches the searchKey without modifying the tree. Returns null or throws exception if no nodes matching are present.
Searching/Retreiving
Retreive(bst, searchKey) { if (bst.empty()) return null; else return root.retreiveNode(searchKey); } retreiveNode(searchKey) { if (searchKey == curRecord) return this; else if (searchKey < curRecord) if (left==null) return null; else return left.retreiveNode(searchKey); else if (right == null) return null; else return right.retreiveNode(searchKey); }
Search Analysis
O(logn) because each decision causes one level to be
descended. There can be at most lg(n) levels to a well balanced tree. Thus at most lg(n) decisions are made before the result is determined.
10
7 3 1 5 8 15
20 30 35
Inserting
As easy as insertion because you simply attach a shiny new leaf somewhere at the bottom of the tree... Insert(newRecord) { newLeaf = createLeafTree(newRecord); // the shiny new leaf ...