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Trees Chapter 19, 20 Instructor: Scott Kristjanson CMPT 125/125 SFU Burnaby, Fall 2013 Scope 2 Trees : Trees as data structures Tree terminology Tree implementations Analyzing tree efficiency Tree traversals Expression

  1. Trees Chapter 19, 20 Instructor: Scott Kristjanson CMPT 125/125 SFU Burnaby, Fall 2013

  2. Scope 2 Trees :  Trees as data structures  Tree terminology  Tree implementations  Analyzing tree efficiency  Tree traversals  Expression trees Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 2 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  3. Trees 3 A tree is a non-linear structure in which elements are organized into a hierarchy A tree is comprised of a set of nodes in which elements are stored and edges connect one node to another Each node is located on a particular level There is only one root node in the tree Level 0 Root Edges Root Level 1 Node Node Node Nodes Node Node Level 2 Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 3 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  4. Trees 4 Nodes at the lower level of a tree are the children of nodes at the previous level A node can have only one parent , but may have multiple children Nodes that have the same parent are siblings The root is the only node which has no parent Level 0 Root Node A is the Parent of Nodes D and E Level 1 Node A Node B Node C Children of Node A Siblings Node D Node E Level 2 Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 4 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  5. Trees 5 A node that has no children is a leaf node A node that is not the root and has at least one child is an internal node A subtree is a tree structure that makes up part of another tree We can follow a path through a tree from parent to child, starting at the root A node is an ancestor of a node if it is above it on the path from the root. Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 5 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  6. Trees 6 Nodes that can be reached by following a path from a particular node are the descendants of that node The level of a node is the length of the path from the root to the node The path length is the number of edges to get from the root to the node The height of a tree is the length of the longest path from the root to a leaf Level 0 Root Height = 2 Node A Node B Node C Level 1 Node D Node E Level 2 Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 6 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  7. Trees – Quiz 7 What are the descendents of node B? What is the level of node E? What is the path length to get from the root to node G? What is the height of this tree? Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 7 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  8. Classifying Trees 8 Trees can be classified in many ways One important criterion is the maximum number of children any node in the tree may have This may be referred to as the order of the tree General trees have no limit to the number of children a node may have A tree that limits each node to no more than n children is referred to as an n-ary tree The tree for play TicTacToe is an 9-ary tree (at most 9 moves) Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 8 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  9. Binary Trees 9 Trees in which nodes may have at most two children are called binary trees Root Node A Node B Node E Node F Node C Node D Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 9 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  10. Balanced Trees 10 A tree is balanced if all of the leaves of the tree are on the same level or within one level of each other Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 10 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  11. Full and Complete Trees 11 A balanced n-ary tree with m elements has a height of log n m A balanced binary tree with n nodes has a height of log 2 n An n-ary tree is full if all leaves of the tree are at the same height and every non-leaf node has exactly n children A tree is complete if it is full, or full to the next-to-last level with all leaves at the bottom level on the left side of the tree Full Binary Tree Complete Binary Tree Root Root Node A Node B Node A Node B Node E Node F Node E Node C Node D Node C Node D Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 11 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  12. Full and Complete Trees 12 Three complete trees: Which trees are full? Only tree c is full Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 12 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  13. Implementing Trees 13 An obvious choice for implementing trees is a linked structure Array-based implementations are the less obvious choice, but are sometimes useful Let's first look at the strategy behind two array-based implementations Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 13 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  14. Computed Child Links 14 For full or complete trees, can use an array to represent a tree For a binary tree with any element stored in position n, • the element’s left child is stored in array position (2n+1) • the element’s right child is stored in array position (2*(n+1)) If the represented tree is not complete or relatively complete, this approach can waste large amounts of array space Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 14 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  15. Simulated Child Links 15 Each element of the array is an object that stores a reference to the tree element and the array index of each child This approach is modeled after the way operating systems manage memory Array positions are allocated on a first-come, first-served basis Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 15 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  16. Tree Traversals 16 For linear structures, the process of iterating through the elements is fairly obvious (forwards or backwards) For non-linear structures like a tree, the possibilities are more interesting Let's look at four classic ways of traversing the nodes of a tree All traversals start at the root of the tree Each node can be thought of as the root of a subtree Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 16 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  17. Tree Traversals 17 Preorder : visit the root, then traverse the subtrees from left to right Inorder : traverse the left subtree, then visit the root, then traverse the right subtree Postorder : traverse the subtrees from left to right, then visit the root Level-order : visit each node at each level of the tree from top (root) to bottom and left to right Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 17 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  18. Tree Traversals 18 Preorder: A B D E C Inorder: D B E A C Postorder: D E B C A Level-Order: A B C D E Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 18 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  19. Tree Traversals 19 Recursion simplifies the implementation of tree traversals Preorder (pseudocode): Visit node Traverse (left child) Traverse (right child) Inorder: Traverse (left child) Visit node Traverse (right child) Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 19 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  20. Tree Traversals 20 Postorder: Traverse (left child) Traverse (right child) Visit node A level-order traversal is more complicated It requires the use of extra structures (such as queues and/or lists) to create the necessary order Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 20 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  21. A Binary Tree ADT 21 Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 21 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  22. A Binary Tree ADT 22 Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 22 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  23. A Binary Tree ADT 23 public interface BinaryTreeADT<T> { public T getRootElement(); public boolean isEmpty(); public int size(); public boolean contains(T targetElement); public T find(T targetElement); public String toString(); public Iterator<T> iterator(); public Iterator<T> iteratorInOrder(); public Iterator<T> iteratorPreOrder(); public Iterator<T> iteratorPostOrder(); public Iterator<T> iteratorLevelOrder(); } Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 23 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

  24. Expression Trees 24 An expression tree is a tree that shows the relationships among operators and operands in an expression An expression tree is evaluated from the bottom up Scott Kristjanson – CMPT 125/126 – SFU Wk12.3 Slide 24 Slides based on Java Foundations 3rd Edition, Lewis/DePasquale/Chase

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