CS : Data Structures

Transcription

1 CS : Data Structures Michael Schatz Oct 12, 2016 Lecture 17: Trees

2 Assignment 5: Due Sunday Oct 10pm Remember: javac Xlint:all & checkstyle *.java & JUnit Solutions should be independently written!

3 Midterm Review

4 Part 2: Trees

5 Trees are all around us J

6 Types of Trees Unordered Binary tree Linear List 3-ary Tree (k-ary tree has k children) Single root node (no parent) Each internal node has at most 1 parent Node may have 0 or more children

7 Not Trees Node 4 has 2 parents Forms a (self) cycle 2 root nodes: Forest Single root node (no parent) Each internal node has at most 1 parent Node may have 0 or more children

8 Special Trees Height of root = 0 Total Height = 3 Full Binary Tree Every node has 0 or 2 children Complete Binary Tree Every level full, except potentially the bottom level What is the maximum number of leaf nodes in a complete binary tree? What is the maximum number of nodes in a complete binary tree? 2 h+1-1 What fraction of the nodes in a complete binary tree are leaves? about half 2 h

9 Balancing Trees Balanced Binary Tree Minimum possible height Unbalanced Tree Non-minimum height What is the height of a balanced binary tree? What is the height of a balanced 3-ary tree? What is the height of a k-ary tree? How much taller is a binary tree than a k-ary tree? What is the maximum height of an unbalanced tree? lg n log 3 n log k n lg n/log k n = lg k n-1 L

10 Tree Interface public interface Tree<T>{ Position <T> insertroot(t t ) throws TreeNotEmptyException; Why insertroot? Position <T> insertchild(position <T> p, T t) throws InvalidPositionException; boolean empty(); Position <T> root() throws TreeEmptyException; Position <T>[] children(position <T> p) Why children()? throws InvalidPositionException, LeafException; Position <T> parent(position<t> p) throws InvalidPositionException ; boolean leaf (Position <T> p) throws InvalidPositionException ; What should parent(root()) return? } T remove(position<t> p) What should remove(root()) do? throws InvalidPositionException, NotALeafException;

11 Tree Traversals + 1 * 2 3

12 Tree Traversals + 1 * 2 3

13 Tree Traversals + 1 * 2 3 Note here we visit children from left to right, but could go right to left Notice we visit internal nodes 3 times: 1: stepping down from parent 2: after visiting first child 3: after visiting second child Different algs work at different times 1: preorder 2: inorder 3: postorder

14 Tree Traversals + 1 * 2 3 Note here we visit children from left to right, but could go right to left Notice we visit internal nodes 3 times: 1: stepping down from parent 2: after visiting first child 3: after visiting second child Different algs work at different times 1: preorder + 1 * 2 3 2: inorder 3: postorder

15 Tree Traversals + 1 * 2 3 Note here we visit children from left to right, but could go right to left Notice we visit internal nodes 3 times: 1: stepping down from parent 2: after visiting first child 3: after visiting second child Different algs work at different times 1: preorder + 1 * 2 3 2: inorder * 3 3: postorder

16 Tree Traversals + 1 * 2 3 Note here we visit children from left to right, but could go right to left Notice we visit internal nodes 3 times: 1: stepping down from parent 2: after visiting first child 3: after visiting second child Different algs work at different times 1: preorder + 1 * 2 3 2: inorder * 3 3: postorder * +

17 Traversal Implementations How to traverse? B A C D basictraversal(node n): // just entered from top basictraversal(n.left) // just got back from left basictraversal(n.middle) // just got back from middle basictraversal(n.right) // just got back from right return // leave to top E F G H I J K L

18 InOrder Traversals How to inorder print? B A C InOrderTraversal(Node n): if n is not null InOrderTraversal(n.left) print(n) InOrderTraversal(n.right) D E F G H I J K L

19 PostOrder Traversals How to postorder print? A PostOrderTraversal(Node n): for c in x.children: PostOrderTraversal(c) print(n) B C D E F G H I J K L

20 PreOrder Traversals [A] [D,C,B] [D,C,F,E] [D,C,F] [D,C,J] [D,C] [D] [I, H, G] B A C D How to preorder print? PreOrderTraversal(Node n): Stack s s.push(n) while (!s.empty()): Node x = s.pop() print(x) for c in x.children: s.push(c) E F G H I J K L

21 Next Steps 1. Reflect on the magic and power of Trees! 2. Take a break from homework J

22 Welcome to CS Questions?