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**Unformatted text preview: **CS161 - Search Trees David Kauchak • Binary Search - Given a sorted list of values A , find a particular value. Similar to looking something up in a dictionary or phone book: O (log n ) • Binary search tree (BST) - A binary search tree is a binary tree where a parents value is greater than all children to the left and less than or equal to all children to the right. Specifically, given a node x in a BST: Left ( x ) < x ≤ Right ( x ) As with other tree structures, can be implemented with pointers or with an array Look at example(s) – Given the definition, what else can we say? * All elements to the left of a node are less than the node * All elements to the right of a node are greater than or equal to the node * The smallest element is the left-most node * The largest element is the right-most node – Why not the setup below?: Left ( x ) ≤ x ≤ Right ( x ) – Which of the set operations is this data structure good/bad for? * Search ( S,k ) - good * Insert ( S,k ) - average 1 * Delete ( S,x ) - average * Minimum ( S ) - good * Maximum ( S ) - good – Enumerating the elements in order: InorderTreeWalk ( x ) 1 if x negationslash = null 2 InorderTreeWalk ( Left ( x )) 3 print x 4 InorderTreeWalk ( Right ( x )) * Is it correct? Definition of BST: Left ( x ) < x ≤ Right ( x ) and proof by induction. * Runtime? Given a node with k nodes in the left subtree and n- k- 1 nodes in the right subtree, the recurrence is: T ( n ) = T ( k ) + T ( n- k- 1) + c we can solve this, or, answer the following two questions: 1. How much work is done for each call to InorderTreeWalk ? 2. How many calls are made to InorderTreeWalk ? * What needs to be changed to traverse in reverse order? * Pre-order and post-order traversals? – Searching for a particular value: BSTSearch ( x,k ) 1 if x = null or k = x 2 return x 3 elseif k < x 4 return BSTSearch ( Left (x), k) 5 else 6 return BSTSearch ( Right (x), k) 2 IterativeBSTSearch ( x,k ) 1 while x negationslash = null and k negationslash...

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