Lecture 04 Trees

1. Lecture 04 Trees • Topics • Trees • Binary Trees • Binary Search trees Trees

2. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada Trees • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments Trees

3. A C D B E G H F K I J Trees • Subtree: tree consisting of a node and its descendants • Terminology • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree Trees

4.  Trees • A node is represented by an object storing • Element • Parent node • Sequence of children nodes • Node objects implement the Position ADT B   A D F B A D F   C E C E Trees

5. Trees • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update method: • object replace (p, o) • Additional update methods may be defined by data structures implementing the Tree ADT • Operations: • We use positions to abstract nodes • Generic methods: • integer size() • boolean isEmpty() • Iterator elements() • Iterator positions() • Accessor methods: • position root() • position parent(p) • positionIterator children(p) Trees

6. Trees AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) • A traversal visits the nodes of a tree in a systematic manner • In a preorder traversal, a node is visited before its descendants • Application: print a structured document 1 Make Money Fast! 2 5 9 1. Motivations 2. Methods References 6 7 8 3 4 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity Trees

7. Trees AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) • In a postorder traversal, a node is visited after its descendants • Application: compute space used by files in a directory and its subdirectories 9 cs16/ 8 3 7 todo.txt1K homeworks/ programs/ 4 5 6 1 2 Robot.java20K h1c.doc3K h1nc.doc2K DDR.java10K Stocks.java25K Trees

8. Binary Trees • Applications: • arithmetic expressions • decision processes • searching • A binary tree is a tree with the following properties: • Each internal node has at most two children (exactly two for proper binary trees) • The children of a node are an ordered pair • We call the children of an internal node left child and right child • Alternative recursive definition: a binary tree is either • a tree consisting of a single node, or • a tree whose root has an ordered pair of children, each of which is abinary tree A C B D E F G I H Trees

9. D C A B E Binary Trees  • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT   B A D     C E Trees

10. A … B D C E F J G H Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11 Trees

11. +   2 - 3 b a 1 Binary Trees • Arithmetic Expression Tree: Binary tree associated with an arithmetic expression • internal nodes: operators • external nodes: operands • Example: arithmetic expression tree for the expression (2  (a - 1) + (3  b)) Trees

12. Binary Trees • Decision Tree: Binary tree associated with a decision process • internal nodes: questions with yes/no answer • external nodes: decisions • Example: dining decision Want a fast meal? No Yes How about coffee? On expense account? Yes No Yes No Starbucks Spike’s Al Forno Café Paragon Trees

13. Binary Trees • Properties: • e = i +1 • n =2e -1 • h  i • h  (n -1)/2 • e 2h • h log2e • h log2 (n +1)-1 • Notation n number of nodes e number of external nodes i number of internal nodes h height Trees

14. The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT (e.g., postOrder/ preOrder traversals) Additional methods: position left(p) position right(p) boolean hasLeft(p) boolean hasRight(p) Update methods may be defined by data structures implementing the BinaryTree ADT Binary Trees Trees

15. Binary Trees AlgorithminOrder(v) ifhasLeft (v) inOrder (left (v)) visit(v) ifhasRight (v) inOrder (right (v)) • In an inorder traversal a node is visited after its left subtree and before its right subtree 6 2 8 1 4 7 9 3 5 Trees

16. +   2 - 3 b a 1 Binary Trees AlgorithmprintExpression(v) ifhasLeft (v)print(“(’’) inOrder (left(v)) print(v.element ()) ifhasRight (v) inOrder (right(v)) print (“)’’) • Print Arithmetic Expressions: Application and specialization of an inorder traversal • print operand or operator when visiting node • print “(“ before traversing left subtree • print “)“ after traversing right subtree ((2  (a - 1)) + (3  b)) Trees

17. +   2 - 3 2 5 1 Binary Trees AlgorithmevalExpr(v) ifisExternal (v) returnv.element () else x evalExpr(leftChild (v)) y evalExpr(rightChild (v))   v. operatorElement () returnx  y • Evaluate Arithmetic Expressions: Specialization of a postorder traversal • recursive method returning the value of a subtree • when visiting an internal node, combine the values of the subtrees Trees

18. Binary Trees • Euler Tour Traversal: Generic traversal of a binary tree • Includes a special cases the preorder, postorder and inorder traversals • Walk around the tree and visit each node three times: • on the left (preorder) • from below (inorder) • on the right (postorder) +   L R B 2 - 3 2 5 1 Trees

19. Binary Trees public abstract class EulerTour{ protected BinaryTree tree;protected voidvisitExternal(Position p, Result r) { }protected voidvisitLeft(Position p, Result r) { }protected voidvisitBelow(Position p, Result r) { }protected voidvisitRight(Position p, Result r) { }protected Object eulerTour(Position p) { Result r = new Result();if tree.isExternal(p) { visitExternal(p, r); }else {visitLeft(p, r); r.leftResult = eulerTour(tree.left(p));visitBelow(p, r); r.rightResult = eulerTour(tree.right(p)); visitRight(p, r);return r.finalResult; } … • Generic algorithm that can be specialized by redefining certain steps • Implemented by means of an abstract Java class • Visit methods that can be redefined by subclasses • Template method eulerTour • Recursively called on the left and right children • A Result object with fields leftResult, rightResult and finalResult keeps track of the output of the recursive calls to eulerTour Trees

20. Binary Trees public class EvaluateExpressionextends EulerTour{ protected voidvisitExternal(Position p, Result r) {r.finalResult = (Integer) p.element(); } protected voidvisitRight(Position p, Result r) {Operator op = (Operator) p.element();r.finalResult = op.operation( (Integer) r.leftResult, (Integer) r.rightResult ); } … } • We show how to specialize class EulerTour to evaluate an arithmetic expression • Assumptions • External nodes store Integer objects • Internal nodes store Operator objects supporting method operation (Integer, Integer) Trees

21. A binary search tree is a binary tree storing keys (or key-value entries) at its internal nodes and satisfying the following property: Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u)key(v) key(w) External nodes do not store items An inorder traversal of a binary search trees visits the keys inincreasing order 6 2 9 1 4 8 Binary Search Trees Trees

22. Binary Search Trees AlgorithmTreeSearch(k, v) ifT.isExternal (v) returnv if k<key(v) returnTreeSearch(k, T.left(v)) else if k=key(v) returnv else{ k>key(v) } returnTreeSearch(k, T.right(v)) • To search for a key k, we trace a downward path starting at the root • The next node visited depends on the outcome of the comparison of k with the key of the current node • If we reach a leaf, the key is not found and we return nukk • Example: find(4): • Call TreeSearch(4,root) 6 < 2 9 > = 8 1 4 Trees

23. Binary Search Trees - Insertion 6 < • To perform operation inser(k, o), we search for key k (using TreeSearch) • Assume k is not already in the tree, and let let w be the leaf reached by the search • We insert k at node w and expand w into an internal node • Example: insert 5 2 9 > 1 4 8 > w 6 2 9 1 4 8 w 5 Trees

24. Binary Search Trees - Deletion 6 < • To perform operation remove(k), we search for key k • Assume key k is in the tree, and let let v be the node storing k • If node v has a leaf child w, we remove v and w from the tree with operation removeExternal(w), which removes w and its parent • Example: remove 4 2 9 > v 1 4 8 w 5 6 2 9 1 5 8 Trees

25. Binary Search Trees - Deletion (cont.) 1 v • We consider the case where the key k to be removed is stored at a node v whose children are both internal • we find the internal node w that follows v in an inorder traversal • we copy key(w) into node v • we remove node w and its left child z (which must be a leaf) by means of operation removeExternal(z) • Example: remove 3 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9 Trees

26. Binary Search Trees -Performance • Consider a dictionary with n items implemented by means of a binary search tree of height h • the space used is O(n) • methods find, insert and remove take O(h) time • The height h is O(n) in the worst case and O(log n) in the best case Trees