1. Lecture 7: Searching a Tree Neil Ghani University of Strathclyde

2. Recall A Tree is either i) A leaf storing an integer ii) A node storing a left subtree, and integer and a right subtree For example, the following are trees leaf 5 node (leaf 5) 6 (leaf 4)

3. Defining Algorithms on Trees We define algorithms on trees by saying i) What do they do on leaves ii) What do they do on nodes For example, the reflect example ref (leaf x) = leaf x ref (node l x r) = node (ref r) x (ref l)

4. Examples Example: An algorithm to add all the numbers in a tree sum (leaf x) = x sum (node l x r) = sum l + x + sum r Note the use of recursion. ALWAYS consider recursion Example: Write an algorithm that takes a number and a tree as input and cuts the tree at that depth

5. Examples Example: An algorithm that takes a number and a tree as input and cuts the tree at that depth cutAt 0 (leaf x) = leaf x cutAt 0 (node l x r) = leaf x cutAt (n+1) (node l x r) = node (cutAt n l) x (cutAt n r)

6. Searching Trees We saw two algortihms for searching a list * Linear search which has O(n) complexity * Binary search which has O(log n) complxity For trees, there are three linear algorithms * In-order search * Pre-order search * Post-order search

7. Pre-Order Traversal Pre-order traversal looks at the root FIRST * Search the root * Search the left subtree * Search the right subtree Example: Printing using preorder traversal pre (leaf x) = putStr x pre (node l x r) = putStr x; pre l; pre r

8. In-Order Traversal In order traversal looks at the root second * Search the left subtree * Search the root * Search the right subtree Example: Printing using inorder traversal inord (leaf x) = putStr x inord (node l x r) = inord l; putStr x; pre r

9. Post-Order Traversal Post order traversal looks at the root LAST * Search the left subtree * Search the right subtree * Search the root Example: Printing using postorder traversal post (leaf x) = putStr x post (node l x r) = post l; post r; putStr x;

10. Example Traversing the following tree gives 6 / \ 3 5 / \ 4 8 pre:6, 3, 4, 8, 5 in:4, 3, 8, 6, 5 post:4, 8, 3, 5, 6