1 / 23

Chapter 4: Trees

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java. Chapter 4: Trees. General Tree Concepts Binary Trees. Lydia Sinapova, Simpson College. Trees. Definitions Representation Binary trees Traversals Expression trees. Definitions.

arin
Download Presentation

Chapter 4: Trees

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 4: Trees General Tree Concepts Binary Trees Lydia Sinapova, Simpson College

  2. Trees • Definitions • Representation • Binary trees • Traversals • Expression trees

  3. Definitions • tree - a non-empty collection of vertices & edges • vertex (node)- can have a name and carry other associatedinformation • path- list of distinct vertices in which successive vertices are connected by edges

  4. Definitions • any two vertices must have one and only one path between them else its not a tree • a tree withN nodeshasN-1 edges

  5. Definitions • root- starting point (top) of the tree • parent(ancestor) - the vertex “above” this vertex • child (descendent) - the vertices “below” this vertex

  6. Definitions • leaves(terminal nodes) - have no children • level - the number of edges between this node and the root • ordered tree - where children’s order is significant

  7. Definitions • Depth of a node- the length of the path from the root to that node • root: depth 0 • Height of a node- the length of the longest path from that node to a leaf • any leaf: height 0 • Height of a tree:The length of the longest path from the root to a leaf

  8. Balanced Trees • the difference between the height of the left sub-tree and the height of the right sub-tree is not more than 1.

  9. root E A R E Child (of root) S T A Leaves or terminal nodes M P L E Depth of T: 2 Height of T: 1 Trees - Example Level 0 1 2 3

  10. Tree Representation Class TreeNode { Object element; TreeNode firstChild; TreeNode nextSibling; }

  11. Example a b f e a g d c b e f g c d

  12. S P O I N S M A Internal node External node D B N Binary Tree

  13. L 0 L 1 L 2 L 3 Height of a Complete Binary Tree At each level the number of the nodes is doubled. total number of nodes: 1 + 2 + 22 + 23 = 24 - 1 = 15

  14. Nodes and Levels in a Complete Binary Tree Number of the nodes in a tree with M levels: 1 + 2 + 22 + …. 2M = 2 (M+1) - 1 = 2*2M - 1 Let N be the number of the nodes. N = 2*2M - 1, 2*2M = N + 1 2M = (N+1)/2 M = log( (N+1)/2 ) N nodes : log( (N+1)/2 ) = O(log(N)) levels M levels: 2 (M+1) - 1 = O(2M ) nodes

  15. Binary Tree Node Class BinaryNode { Object Element; // the data in the node BinaryNode left; // Left child BinaryNode right; // Right child }

  16. C O T M E R U P L N A D Binary Tree – Preorder Traversal Root Left Right First letter - at the root Last letter – at the rightmost node

  17. Preorder Algorithm preorderVisit(tree) { if (current != null) { process (current); preorderVisit (left_tree); preorderVisit (right_tree); } }

  18. U P T O R E C M A D L N Binary Tree – Inorder Traversal Left Root Right First letter - at the leftmost node Last letter – at the rightmost node

  19. Inorder Algorithm inorderVisit(tree) { if (current != null) { inorderVisit (left_tree); process (current); inorderVisit (right_tree); } }

  20. D P N M A U C O L R T E Binary Tree – Postorder Traversal Left Right Root First letter - at the leftmost node Last letter – at the root

  21. Postorder Algorithm postorderVisit(tree) { if (current != null) { postorderVisit (left_tree); postorderVisit (right_tree); process (current); } }

  22. Expression Trees The stack contains references to tree nodes (bottom is to the left) 3 + * 1 2 2 3 1 + (1+2)*3 2 1 Post-fix notation:1 2 + 3 *

  23. * 3 + 2 1 Expression Trees In-order traversal: (1 + 2) * ( 3) Post-order traversal: 1 2 + 3 *

More Related