Data Structures, Algorithms & Complexity

1. GRIFFITH COLLEGE DUBLIN Data Structures, Algorithms & Complexity Tree Algorithms Lecture 8

2. Introduction • Trees are a mathematical abstraction that play a central role in the design and analysis of algorithms • We use trees to describe dynamic properties of algorithms • We build and use explicit data structures that are concrete realisations of trees • Definition of Trees • Tree • Rooted tree • Ordered tree • M-ary tree and binary tree Lecture 8

3. Tree • A tree is a non-empty collection of vertices and edges that satisfies certain requirements. • A vertex (or node) is a simple object that can have a name and can carry other associated information • An edge is a connection between two vertices • A path in a tree is a list of distinct vertices in which successive vertices are connected by edges in the tree • The defining property of a tree is that there is precisely one path connecting any two nodes. • A disjoint set of trees is called a forest Lecture 8

4. Rooted Tree • A Rooted Tree is one where we designate one node as the root • In computer science we normally reserve the term tree to refer to rooted trees. The more general structure is a free tree • In a rooted tree, any node is the root of a subtree consisting of it and the nodes below it • There is exactly one path between the root and each of the other nodes in the tree • Each node except the root has exactly one node above it in the tree, called its parent, and we extend the family analogy talking of children, siblings, or grandparents • Nodes with no children are leaves or terminal nodes Lecture 8

5. Ordered Tree • An ordered tree is a rooted tree in which the order of the children at every node is specified. • If each node must have a specific number of children appearing in a specific order, then we have an M-ary tree • The simplest type of M-ary tree is the binary tree • A binary tree is an ordered tree consisting of nodes that can have at most two children • As with any Abstract Data Structure we can implement a binary tree in a number of ways, using arrays, strings, or structures and pointers Lecture 8

6. Examples of Trees • A free tree A rooted binary tree A forest Lecture 8

7. Mathematical Properties • Nodes on a tree are internal or external • An internal node is a node in the tree • An external node is a node that could be added to the existing nodes • A binary tree with N internal nodes has N+1 external nodes • A binary tree with N internal nodes has 2N links • N-1 links to internal nodes and N+1 links to external nodes internal nodes external nodes Lecture 8

8. Terminology • The level of a node in a tree is one higher than the level of its parent (with the root at level 0) • The height of a tree is the maximum of the levels of the tree’s nodes • The height of a complete binary tree with N internal nodes is lg N • The path length of a tree is the sum of the levels of all the trees nodes Lecture 8

9. Tree Traversal • Given a tree we want to process each node in the tree systematically • For binary trees, we have two links, and we therefore have three basic orders in which we might visit the nodes: • Preorder, where we visit the node, then visit the left and right subtrees • Inorder, where we visit the left subtree, then visit the node, then visit the right subtree • Postorder, where we visit the left and right subtrees, then visit the node • These can be implemented easily using a recursive implementation Lecture 8

10. Pre-Order Traversal Preorder(T) if (T <> nil) then visit(T) Preorder(left(T)) Preorder(right(T)) endif endalg • Assuming that ‘visit’ means print and we have the following tree: 5 3 7 2 5 8 • The output of the call Preorder(T) will be, 5 3 2 5 7 8 Lecture 8

11. Post and In order • Changing to Postorder or Inorder simply involves changing the order of the calls • Preordervisits the root before calling the function for either subtree • Postordervisits the root after calling the function for the subtrees • Inorder calls the function for the left subtree, then visits the root, and then calls the function for the right subtree Lecture 8

12. Postorder and Inorder Algorithms Postorder(T) Inorder(T) if (T <> nil) then if (T <> nil) then Postorder(left(T)) Inorder(left(T)) Postorder(right(T)) visit(T) visit(T) Inorder(right(T)) endif endif endalg endalg Lecture 8

13. Traversal Example • Given the following tree above the traversals will give output as shown E D H B F A C G • Preorder : E D B A C H F G • Inorder : A B C D E F G H • Postorder: A C B D G F H E Lecture 8

14. Binary Search Tree • A binary search tree is a binary tree in which every item in the left subtree is less than or equal to the item at the root, and every item in the right subtree is greater than the item in the root. 5 3 7 2 5 8 • An inorder traversal of a binary search tree will print the nodes in sorted order • The most common operation performed on a binary search tree is searching for a key. Lecture 8

15. Binary Search Tree Algorithms • Let x be a reference to the root of the tree, and k be the key we are looking for Tree-Search(x, k) ifx = nilork = key[x] then returnx endif ifk < key[x] then return Tree-Search( left[x], k) else return Tree-Search( right[x], k) endif endalg Lecture 8

16. Binary Search Tree Algorithms • Tree Minimum and Maximum are simple procedures on a binary search tree Tree-Minimum(x)Tree-Maximum(x) while left[x] <> nil while right[x] <> nil x = left[x] x = right[x] endwhile endwhile returnx return x endalg endalg Lecture 8

17. Summary • Trees are a mathematical abstraction that play a central role in the design and analysis of algorithms • We build and use explicit data structures that are concrete realisations of trees • Trees can be free, rooted, ordered or M-ary • Trees have height, level and path length, all of which are defined • Tree Traversal is an important operation • For a binary tree we looked a preorder, postorder and inorder traversal • Introduced a binary search tree and some simple algorithms. Will return to this. Lecture 8