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Understanding Binary Trees and Their Traversals

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randi
23 diapositives · 353 Views · Jul 23, 2014

This chapter introduces the fundamental concepts of binary trees, including their structure, properties, and traversal methods. Binary trees are defined recursively, where each node can link to two or more child nodes, with leaves having no children. Key concepts such as depth, height, size, and the role of root nodes are covered, along with practical applications in computing systems, like file structures and indexing. Various tree traversal methods (preorder, inorder, postorder) are introduced, emphasizing recursive approaches and stack usage for maintaining traversal states.

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Understanding Binary Trees and Their Traversals
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  1. Chapter18- basic definitions- binary trees- tree traversals Intro. to Trees CSCI 3333 Data Structures

  2. Trees • A recursively defined data structure • Each node has links to two or more subtrees/children-nodes. • Each node itself can be the beginning of a new tree. • Nodes with no children nodes are called leaves or leaf nodes. • The root node has no parent. • Example class BinaryNode<AnyType> { private AnyType element; private BinaryNode<AnyType> left; private BinaryNode<AnyType> right; … }

  3. Trees • Depth of a node: the length of the path from the root to the node. • Height of a node: the length of the path from the node to the deepest leaf. • Nodes in a tree are linked by edges. • Sibling nodes: nodes with the same parent. • Ancestor nodes vs descendent nodes • The size of a node is the number of descendents plus itself. CSCI 3333 Data Structures

  4. More Tree Properties • The depth of any node is 1 more than the depth of its parent. • The height of any node is 1 more than the height of its maximum-height child. • The height of a tree is the height of the root node. • The size of a tree is the size of its root. • A tree with N nodes has N-1 edges. (Except for the root, each of the other nodes has one incoming edge.) CSCI 3333 Data Structures

  5. Implementation of Trees • Each node is allocated space for data plus a number of references to other nodes (children, siblings). • Exercise: Define the tree shown below as a class. • Q: What’s the space allocation for each of the nodes in the following tree? CSCI 3333 Data Structures

  6. Implementation of Trees • An alternative implementation • Exercise: Define a class to represent such a tree. • Q: What’s the space allocation for each of the nodes in this tree? CSCI 3333 Data Structures

  7. Applications of Trees • Many applications in computing systems • Typically a tree is used as an index to the stored data. • Examples: • The file system directory/folder structure • a secondary index to a primary file • An index to a database • etc. CSCI 3333 Data Structures

  8. Binary Trees • A tree in which no node has more than two children. • Many applications: • expression trees (in compiler design), • Huffman coding tree (in data compression), • binary search tree • Priority queues CSCI 3333 Data Structures

  9. BinaryNode.java CSCI 3333 Data Structures

  10. BinaryTree.java CSCI 3333 Data Structures

  11. CSCI 3333 Data Structures

  12. Recursion and Trees • Most methods in trees are defined as recursive methods. • Example: find the size of a tree CSCI 3333 Data Structures

  13. Recursion and Trees • Example: find the height of a tree CSCI 3333 Data Structures

  14. Tree Traversal • Preorder traversal: the node itself + the left child + the right child • Postorder traversal: the left child + the right child + the node • Inorder traversal: the left child + the node + the right child CSCI 3333 Data Structures

  15. CSCI 3333 Data Structures

  16. Tree Traversals and Stacks • The task: Use a stack to maintain the states of a tree when traversing the nodes • Example: postorder traversal Each node in the stack is associated with one of the three counters to represent its state: • If we are about to process the node’s left child • If we are about to process the node’s right child • If we are about to process the node itself Default counter value = 0; The counter value represents the number of times popped. CSCI 3333 Data Structures

  17. Example program: TestTreeIterators.javahttp://users.cis.fiu.edu/~weiss/dsj4/code/TestTreeIterators.java public static <AnyType> void testItr( String type, TreeIterator<AnyType> itr ) { try { System.out.print( type + ":" ); for( itr.first( ); itr.isValid( ); itr.advance( ) ) System.out.print( " " + itr.retrieve( ) ); System.out.println( ); itr.advance( ); } catch( NoSuchElementException e ) { System.out.println( e + " (as expected)" ); } } CSCI 3333 Data Structures

  18. CSCI 3333 Data Structures

  19. CSCI 3333 Data Structures

  20. CSCI 3333 Data Structures

  21. Tree Traversals using Stacks • Example: postordertraversal CSCI 3333 Data Structures

  22. Tree Traversals using Stacks • Exercise 18.4: Show the stack operations when a preorder traversal is applied to the tree shown in Figure 18.25. Inorder traversal ? Lab 5. CSCI 3333 Data Structures

  23. Level-order traversal • Nodes in the tree are processed starting at the root and going from top to bottom, left to right. • Level 0 nodes • Level 1 nodes • Level 2 nodes • … • Exercise: What are the order of printed data if the following trees are level-order traversed? CSCI 3333 Data Structures