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

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## Lecture 04 Trees

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**Lecture 04 Trees**• Topics • Trees • Binary Trees • Binary Search trees Trees**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**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****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**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**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**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**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**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**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**+** 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**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**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**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**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**+** 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**+** 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**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**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**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**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**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**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**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**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**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

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