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Chapter 6 Trees

Chapter 6 Trees. Outline. Basics N-ary Trees Binary Trees Tree Traversals Expression Trees Implementing Trees Tree Traversals Tree Enumerations General Trees N-ary Trees Binary Trees Binary Tree Traversals Comparing Trees Applications. L ec 1. President.

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Chapter 6 Trees

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  1. Chapter 6Trees ICS 202  2011 spring  Data Structures and Algorithms 

  2. Outline • Basics • N-ary Trees • Binary Trees • Tree Traversals • Expression Trees • Implementing Trees • Tree Traversals • Tree Enumerations • General Trees • N-ary Trees • Binary Trees • Binary Tree Traversals • Comparing Trees • Applications

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  4. President Vice president division A Vice president division B Manager dept. A1 Manager dept. A2 Manager dept. B1 Clerk Clerk Clerk Clerk Clerk Clerk 1. Basics • A tree is anon-linear data structure • A tree is often used to represent a hierarchy

  5. 1. Basics Definition 1 A tree is a finite, non-empty set of nodes, with the following properties A chosen node of the set, r, is called the root of the tree; The remaining nodes are subsets, T1, T2, …, Tn, each of each is a tree.

  6. 1. Basics D B G E A F H J M C I K L

  7. Consider a tree 1. Basics Terminology: • The degree of a node is the number of subtrees associated with that node • A node of degree zero has no subtrees. Such a node is called a leaf. • Each root ri of subtree Ti of tree T is called a child of r. • The root node rof tree T is the parent of all the roots riof the subtrees Ti, 1 < i n. The term grandparent. • Two roots ri and rjof distinct subtrees Ti and Tj of tree T are called siblings(brothers). ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  8. Definition 2 (Path and Path Length) • Given a tree T containing the set of nodes R, a path in T is defined as a non-empty sequence of nodes 1. Basics • The length of path P is k-1. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  9. 1. Basics D G E F H J M I K L ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  10. 1. Basics D D G E F H J M E G F H J M I K L I K L ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  11. D G E F H J M I K L 1. Basics class D { class E { class F { } } class G { class H { class I { } } class J { class K { } class L { } class M { } } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  12. 2. N-ary Trees • Definition 3 (N-ary Tree) An N-ary tree T is a finite set of nodes with the following properties: • Either the set is empty, T =; or • The set consists of a root, R, and exactly N distinct N-ary trees. That is, the remaining nodes are partitioned into N 0 subsets, T0, T1, …, TN-1, each of which is an N-ary tree such that T = {R, T0, T1, …, TN-1}. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  13. 1. Basics A A Tertiary tree (N = 3) ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  14. 1. Basics B C B C ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  15. 2. N-ary Trees • The empty trees are called external nodes (they have no subtrees and they appear at the extremities of the tree (squares)). • The non-empty trees are called internal nodes (circles). • A tree can be ordered (if its subtrees are ordered) or oriented if the order of the subtrees doesn’t matter. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  16. Theorem 2 Consider an N-ary tree of height h  0. The maximum number of internal nodes in T is given by: 2. N-ary Trees Theorem 1 An N-ary tree with n 0 internal nodes contains (N-1)n + 1 external The height of a node ri in a tree is the length of the longest path from node ri to a leaf. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  17. Theorem 3 Consider an N-ary tree of height h  0. The maximum number of leaf nodes T is 2. N-ary Trees ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  18. 3. Binary Trees • Definition 4 (Binary Tree) A binary tree T is a finite set of nodes with the following properties: • Either the set is empty, T =; or • The set consists of a root, r, and exactly two distinct binary trees TL, and TR, T = {r, TL, TR}. • TL is called the left subtree of T, and the tree TR is called the right subtree of T. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  19. 3. Binary Trees A A B B • Binary trees are almost considered to be ordered trees (the above two binary trees are different). ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  20. 4. Tree Traversals • There are many different algorithms for manipulating trees. • A most common characteristic of these algorithms their visit of all nodes in the tree (called a tree traversal). • There are essentially two different traversal ways: depth-first traversal and breadth-first traversal. • Some depth-first traversals can be preorder, inorder, or, postorder traversals. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  21. 4. Tree Traversals A depth = 0 B D depth = 1 C E H depth = 2 F G I depth = 3 ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  22. 4. Tree Traversals Preorder Traversal • The first depth-first traversal method is called preorder traversal. • Preorder traversal is defined as follows • Visit the root first, and then • Do a preorder traversal each of the subtrees of the root one-by-one in the order given. • In the case of a binary tree, the algorithm becomes • Visit the root first, and then • Traverse the left subtree, and then • Traverse the right subtree ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  23. A depth = 0 B D depth = 1 C E H depth = 2 F G I depth = 3 4. Tree Traversals Preorder traversal A, B, C, D, E, F, G, H, I ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  24. 4. Tree Traversals Postorder Traversal • The second depth-first traversal method is called postorder traversal. • Postorder traversal is defined as follows • Do a postorder traversal each of the subtrees of the root one-by-one in the order given, and then • Visit the root. • In the case of a binary tree, the algorithm becomes • Traverse the left subtree, and then • Traverse the right, and then • Visit the root. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  25. A depth = 0 B D depth = 1 C E H depth = 2 F G I depth = 3 4. Tree Traversals Postorder traversal C, B, F, G, E, I, H, D, A ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  26. 4. Tree Traversals Inorder Traversal • The third depth-first traversal method is called inorder traversal (available only for binary trees). • Inorder traversal is defined as follows • Traverse the left subtree, and then • Visit the root, and then • Traverse the right subtree. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  27. A depth = 0 B D depth = 1 C E H depth = 2 F G I depth = 3 4. Tree Traversals Inorder Traversal B, C, A, F, E, G, D, I, H ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  28. 4. Tree Traversals Breadth-First Traversal • The Breadth-first traversals are non-recursive traversals and better understood. • A Breadth-first traversal visits the nodes of a tree in the order of their depth. • It visits first nodes at depth zero, then all nodes at depth one, and so on. • At each depth, the nodes are visited from left to right. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  29. A depth = 0 B D depth = 1 C E H depth = 2 F G I depth = 3 4. Tree Traversals Inorder Traversal A, B, D, C, E, H, F, G, I ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

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  31. +   a b - e c d 5. Expression Trees • Leaves = variables (a, b, …) • Non-terminal nodes = operators (+, -, ,) • The non-terminal nodes have either one or two non-empty subtrees (binary or unary operator). ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  32. +   a b - e c d 5. Expression Trees • Inorder traversal of the expression tree: a, , b, +, c, -, d, , e • Except the missing parentheses, the order in which the symbols appear is the same as in the expression. ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  33. 5. Expression Trees • To apply an inorder traversal and take into account the level of operators, we process as following: • When the inorder traversal encounters(found) a terminal node (variable or constant), prints it out • When it encounters a non-terminal node, it does the following: • Print a left parenthesis, and then • Traverse the left subtree, and then • Print the root, and then • Traverse the right subtree, and then • Print a right parenthesis ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  34. +   a b - e c d 5. Expression Trees ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  35. 6. Implementing Trees AbstractObject AbstractContainer Comparable GeneralTree Container Tree NaryTree AbstractTree BinaryTree ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  36. 6. Implementing Trees public interface Tree extends Container { Object getKey (); // returns the object contained in the root of the tree Tree getSubtree (int i); // returns the ith subtree of the tree boolean isEmpty (); // returns true if the root is empty boolean isLeaf (); // returns true if the root of the tree is a leaf int getDegree (); // returns the degree of the root int getHeight (); // returns the height of the tree void depthFirstTraversal (PrePostVisitor visitor); void breadthFirstTraversal (Visitor visitor); } // all the nodes are systematically visited either in depth-first or breadth-first traversal ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  37. 6. Implementing Trees: Tree traversal (depth-first) public abstract class AbstractTree extends AbstractContainer implements Tree { public void depthFirstTraversal (PrePostVisitor visitor) { if (visitor.isDone ()) return; if (!isEmpty ()) { visitor.preVisit (getKey ()); // visit the root for (int i = 0; i < getDegree (); ++i) // visit the nodes under the root getSubtree (i).depthFirstTraversal (visitor); // recursive visitor.postVisit (getKey ()); // visit subtree } } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  38. 6. Implementing Trees: Tree traversal (depth-first) public interface PrePostVisitor { void preVisit (Object object); void inVisit (Object object); void postVisit (Object object); boolean isDone (); } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  39. 6. Implementing Trees: Tree traversal (depth-first) public abstract class AbstractPrePostVisitor implements PrePostVisitor { public void preVisit (Object object) {} public void inVisit (Object object) {} public void postVisit (Object object) {} public boolean isDone () { return false; } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  40. 6. Implementing Trees: Tree traversal (depth-first) public class PreOrder extends AbstractPrePostVisitor { // using the visitor interface protected Visitor visitor; public PreOrder (Visitor visitor) { this.visitor = visitor; } public void preVisit (Object object) { visitor.visit (object); } public boolean isDone () { return visitor.isDone (); } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  41. 6. Implementing Trees: Tree traversal (depth-first) public class InOrder extends AbstractPrePostVisitor { protected Visitor visitor; public InOrder (Visitor visitor) { this.visitor = visitor; } public void inVisit (Object object) { visitor.visit (object); } public boolean isDone () { return visitor.isDone (); } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  42. 6. Implementing Trees: Tree traversal (depth-first) public class PostOrder extends AbstractPrePostVisitor { protected Visitor visitor; public PostOrder (Visitor visitor) { this.visitor = visitor; } public void postVisit (Object object) { visitor.visit (object); } publicboolean isDone () { return visitor.isDone (); } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  43. 6. Implementing Trees: Tree traversal (depth-first) Visistor v = new PrintingVisitor (); Tree t = new SomeTree (); // … t.depthFirstTraversal (new PreOrder (v)); t.depthFirstTraversal (new InOrder (v)); t.depthFirstTraversal (new PostOrder (v)); ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  44. 6. Implementing Trees: Tree traversal (Breadth-first) public abstract class AbstractTree extends AbstractContainer implements Tree { public void breadthFirstTraversal (Visitor visitor) { Queue queue = new QueueAsLinkedList (); if (!isEmpty ()) queue.enqueue (this); // the root node is enqueuedfirstly while (!queue.isEmpty () && !visitor.isDone ()) { Tree head = (Tree) queue.dequeue (); // remove the head visitor.visit (head.getKey ()); // visit the object in head for (int i = 0; i < head.getDegree (); ++i) { Tree child = head.getSubtree (i); if (!child.isEmpty ()) queue.enqueue (child); // enqueue in order each // non-empty subtree } } } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  45. 6. Implementing Trees: Binary Trees A B D C E H F G I ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  46. A  C B D H E    G I       F 6. Implementing Trees: Binary Trees ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  47. 6. Implementing Trees: Binary Trees public class BinaryTree extends AbstractTree { protected Object key; protected BinaryTree left; protected BinaryTree right; // ... } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  48. 6. Implementing Trees: Binary Trees public class BinaryTree extends AbstractTree { protected Object key; protected BinaryTree left; protected BinaryTree right; public BinaryTree (Object key, BinaryTree left, BinaryTree right) { this.key = key; this.left = left; this.right = right; } public BinaryTree () { this (null, null, null); } public BinaryTree (Object key) { this (key, new BinaryTree (), new BinaryTree ()); } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  49. 6. Implementing Trees: Binary Trees public class BinaryTree extends AbstractTree { protected Object key; protected BinaryTree left; protected BinaryTree right; publicvoid purge () { key = null; left = null; right = null; } // ... } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

  50. 6. Implementing Trees: Binary Trees public class BinaryTree extends AbstractTree { protected Object key; protected BinaryTree left; protected BinaryTree right; publicvoid depthFirstTraversal (PrePostVisitor visitor) { if (!isEmpty ()) { visitor.preVisit (key); left.depthFirstTraversal (visitor); visitor.inVisit (key); right.depthFirstTraversal (visitor); visitor.postVisit (key); } } } ICS 202  2007-2  Data Structures and Algorithms  Dr. Youssef Harrath

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