1. CS 261 – Winter 2010 Trees

2. Ubiquitous – they are everywhere in CS Probably ranks third among the most used data structure: 1.Vectors and Arrays 2.Lists 3.Trees

3. Tree Characteristics A tree consists of a collection of nodes connected by directed arcs A tree has a single root node –By convention, the root node is usually drawn at the top A node that points to (one or more) other nodes is the parent of those nodes while the nodes pointed to are the children Every node (except the root) has exactly one parent Nodes with no children are leaf nodes Nodes with children are interior nodes

4. Tree Characteristics (cont.) Nodes that have the same parent are siblings The descendents of a node consist of its children, and their children, and so on –All nodes in a tree are descendents of the root node (except, of course, the root node itself) Any node can be considered the root of a subtree –Like a subset, a subtree need not be “proper” (i.e., be strictly smaller than the original) A subtree rooted at a node consists of that node and all of its descendents

5. Tree Characteristics (cont.) There is a single, unique path from the root to any node –Arcs don’t join together A path’s length is equal to the number of arcs traversed A node’s height is equal to the maximum path length from that node to a leaf node: –A leaf node has a height of 0 –The height of a tree is equal to the height of the root A node’s depth is equal to the path length from the root to that node: –The root node has a depth of 0 –A tree’s depth is the maximum depth of all its leaf nodes (which, of course, is equal to the tree’s height)

6. Tree Characteristics (cont.) Root (depth = 0, height = 4) A BC DE Subtree rooted at node C Nodes D and E are children of node B Node B is the parent of nodes D and E Nodes B, D, and E are descendents of node A (as are all other nodes in the tree…except A) E is an interior node F is a leaf node F Leaf node (depth = 4, height = 0)

7. Tree Characteristics (cont.) Are these trees? YesNo

8. Binary Tree Nodes have no more than two children: –Children are generally ordered from left to right Full Binary Tree: every leaf is at the same depth –Every internal node has 2 children –Height of n will have 2 n+1 – 1 nodes –Height of n will have 2 n leaves

9. Binary Tree Nodes have no more than two children: –Children are generally ordered from left to right Full Binary Tree: every leaf is at the same depth –Every internal node has 2 children –Height of n will have 2 n+1 – 1 nodes –Height of n will have 2 n leaves Complete Binary Tree: full except for the bottom level which is filled from left to right

10. Relationship of Height to Number of Nodes If a complete binary tree has N nodes, what it its height? We will come back to this when We later have algorithms That run in time Proportional to the Path length

11. Array Implementation Complete binary tree have structure that is efficiently implemented with an array : –Children of node i are stored at 2i + 1 and 2i + 2 –Parent of node i is at floor((i - 1) / 2) a bc def 0a0a 1b1b 2c2c 3d3d 4e4e 5f5f 67 Root

12. Array Implementation (cont.) If the tree is not complete (it is thin, unbalanced, etc.), the Array implementation will be full of holes a bc f de 0a0a 1b1b 2c2c 34d4d 56e6e 78910111213 f 1415

13. Dynamic Memory Implementation struct node { EleType value; struct node * left; // Left child. struct node * right; // Right child. }; Like the Link class in LinkedList : we will use this class in several data structures

14. Binary Tree Application: Animal Game Purpose: guess an animal using a sequence of questions –Internal nodes contain yes/no questions –Leaf nodes are animals –Initially, tree contains a single animal (e.g., a “cat”) stored in the root node 1.Start at root. 2.If internal node  ask yes/no question Yes  go to left child and repeat step 2 No  go to right child and repeat step 2 3.If leaf node  ask “I know. Is it a …”: If right  done If wrong  “learn” new animal by asking for a yes/no question that distinguishes the new animal from the guess

15. Binary Tree Traversals Just as a list, it is often necessary to examine every node in a tree A list is a simple linear structure: can be traversed either forward or backward – but usually forward What order do we visit nodes in a tree? Most common traversal orders: –Pre-order –In-order –Post-order

16. Binary Tree Traversals (cont.) All traversal algorithms have to: –Process node –Process left subtree –Process right subtree Traversal order determined by the order these operations are done. Six possible traversal orders: 1.Node, left, right  Pre-order 2.Left, node, right  In-order 3.Left, right, node  Post-order 4.Node, right, left 5.Right, node, left 6.Right, left, node Subtrees are not  usually analyzed  from right to left.  Most common traversals. 

17. Process order  Node, Left subtree, Right subtree // Not in the BinaryNode class. void preorder(BinaryNode node) { if (node != null){ process (node.obj); preorder(node.left); preorder(node.rght); } Example result: p s a m a e l r t e e e l s a amr t e e Pre-Order Traversal p

18. Post-Order Traversal Process order  Left subtree, Right subtree, Node void postorder(BinaryNode node) { if (node != null){ postorder(node.left); postorder(node.rght); process (node.obj); } Example result: a a m s l t e e r e p e l s a mr t e e a p

19. In-Order Traversal Process order  Left subtree, Node, Right subtree void inorder(BinaryNode node) { if (node != null){ inorder(node.left); process (node.obj); inorder(node.rght); } Example result: a sample tree e l s a mr t p a e e

20. Binary Tree Traversals: Euler Tour An Euler Tour “walks” around the tree’s perimeter Each node is visited three times: –1 st visit: left side of node –2 nd visit: bottom side of node –3 rd visit: right side of node –Leaf nodes are visited three times in succession Traversal order depends on when node processed: –Pre-order: 1 st visit –In-order: 2 nd visit –Post-order: 3 rd visit e l s a mr t p a e e

21. Pre-order: + a * + b c d (Polish notation) In-order: a + (b + c) * d (parenthesis added) Post-order: a b c + d * + (reverse Polish notation) * + a d b c Traversal Example +

22. Traversals Computational complexity: –Each traversal requires constant work at each node (not including the recursive calls) –Order not important –Iterating over all n elements in a tree requires O(n) time Problem with traversal code: –The process function must be rewritten (or replaced) for each new task –Programmer writing new process function sees the internal tree structure representation –Not good information hiding –Solution  Iterator (more on this later)

23. Questions? Next topic, how do we make a useful data structure out of a tree?