Sequences 11/12/2007 2:12 PM What is a Tree In computer science, a tree is an abstract model Computers”R”Us of a hierarchical structure A tree consists of nodes Sales Manufacturing R&D with a parent-child relation (at most one parent!) US International Laptops Desktops Applications: Trees Make Money Fast! Stock Fraud Organization charts File systems Europe Programming environments Bank Robbery Ponzi Scheme Trees © 2004 Goodrich, Tamassia 1 Subtree: tree consisting of a node and its descendants B C D G H subtree J 1. Motivations 3 4 © 2004 Goodrich, Tamassia integer size() boolean isEmpty() Iterator iterator() Iterator positions() Update method: 6 Algorithm preOrder(v) visit(v) for each child w of v preorder (w) position root() position parent(p) Iterator children(p) Trees © 2004 Goodrich, Tamassia Trees 4 In a postorder traversal, a node is visited after its descendants Application: compute space used by files in a directory and its subdirectories 9 Algorithm postOrder(v) for each child w of v postOrder (w) visit(v) cs16/ 3 9 7 2.1 Stock Fraud object replace (p, o) Additional update methods may be defined by data structures implementing the Tree ADT Postorder Traversal 2. Methods 1.2 Avidity K 5 3 boolean isInternal(p) boolean isExternal(p) boolean isRoot(p) Make Money Fast! 2 1.1 Greed Query methods: Accessor methods: F Preorder Traversal 1 We use positions to abstract nodes Generic methods: A Trees 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 2 Tree ADT 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 (not counting itself) E Height of a tree: maximum depth of any node (3) Descendant of a node: child, I grandchild, grand-grandchild, etc. © 2004 Goodrich, Tamassia Canada Trees © 2004 Goodrich, Tamassia Tree Terminology Asia 2.2 Ponzi Scheme References 1 2 2.3 Bank Robbery h1c.doc 3K h1nc.doc 2K © 2004 Goodrich, Tamassia todo.txt 1K programs/ 8 5 8 7 homeworks/ 4 5 DDR.java 10K Trees Stocks.java 25K 6 Robot.java 20K 6 1 Sequences 11/12/2007 2:12 PM Binary Trees Arithmetic Expression Tree B a tree consisting of a single node, or a tree whose root has an ordered pair of children, each of which is a root of a binary tree Example: arithmetic expression tree for the expression (2 × (a − 1) + (3 × b)) + D E × F 7 The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT Additional methods: internal nodes: questions with yes/no answer external nodes: decisions Want a fast meal? No How about coffee? On expense account? Yes No Yes No Starbucks Spike’s Al Forno Café Paragon Trees 9 Inorder Traversal In an inorder traversal a node is visited after its left subtree and before its right subtree Application: draw a binary tree 4 3 8 7 Trees 10 Print Arithmetic Expressions Specialization of an inorder traversal print operand or operator when visiting node print “(“ before traversing left subtree print “)“ after traversing right subtree + × 9 × − 2 5 a Trees Update methods may be defined by data structures implementing the BinaryTree ADT position left(p) position right(p) boolean hasLeft(p) boolean hasRight(p) © 2004 Goodrich, Tamassia 8 2 © 2004 Goodrich, Tamassia Algorithm inOrder(v) if hasLeft (v) inOrder (left (v)) visit(v) if hasRight (v) inOrder (right (v)) x(v) = inorder rank of v y(v) = depth of v 6 1 Trees BinaryTree ADT Yes b 1 © 2004 Goodrich, Tamassia Example: dining decision 3 a Trees © 2004 Goodrich, Tamassia − I Binary tree associated with a decision process × 2 G Decision Tree internal nodes: operators external nodes: operands C H © 2004 Goodrich, Tamassia A We call the children of an internal node left child and right child Alternative recursive definition: a binary tree is either arithmetic expressions decision processes searching Each internal node has at most two children (exactly two for proper, or perfectly balanced, binary trees) The children of a node are an ordered pair Binary tree associated with an arithmetic expression Applications: A binary tree is a tree with the following properties: 11 3 Algorithm printExpression(v) if hasLeft (v) print(“(’’) printExpression (left(v)) print(v.element ()) if hasRight (v) printExpression(right(v)) print (“)’’) b ((2 × (a − 1)) + (3 × b)) 1 © 2004 Goodrich, Tamassia Trees 12 2 Sequences 11/12/2007 2:12 PM 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 + × Algorithm evalExpr(v) if isExternal (v) return v.element () else x ← evalExpr(leftChild (v)) y ← evalExpr(rightChild (v)) ◊ ← operator stored at v return x ◊ y × − 2 5 3 13 A D D F F ∅ E ∅ E Trees © 2004 Goodrich, Tamassia 14 Array-Based Representation of Binary Trees nodes are stored in an array ∅ A B ∅ B … 2 ∅ A D ∅ E ∅ ∅ C Trees ∅ 15 4 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 © 2004 Goodrich, Tamassia 5 E Trees 6 7 C F 10 J 11 G H 16 Height is logarithmic in size A binary tree is proper if every node has exactly 2 children, apart from the last level which only has leaves. level 0 Proper level 1 binary tree level 2 of height 3: This is a very important property of proper, or perfectly balanced, binary trees Tree algorithms often work by going down the tree level by level So, their running time depends on the number of levels – but we usually only know the number of nodes Let us prove that h = log2 (n + 1) − 1 where n in the number of nodes. Or (same thing) number of levels = log2 (n + 1). level 3 Trees E D let rank(node) be defined as follows: D 3 B Properties of proper binary trees © 2004 Goodrich, Tamassia ∅ 1 Node objects implement the Position ADT © 2004 Goodrich, Tamassia ∅ C Element Parent node Left child node Right child node C B B C Trees A ∅ Node objects implement the Position ADT 1 A node is represented by an object storing Element Parent node Sequence of children nodes A Linked Structure for Binary Trees A node is represented by an object storing 2 © 2004 Goodrich, Tamassia Linked Structure for Trees 17 © 2004 Goodrich, Tamassia Trees 18 3 Sequences 11/12/2007 2:12 PM How many nodes at level k How many nodes at level k First, it is useful to find out how many nodes are at a certain level in the proper binary tree Let us count levels from 0. This way level k contains nodes which have depth k. © 2004 Goodrich, Tamassia Trees 19 Claim: level k contains 2k nodes. Proof: by induction on k. (basis of induction) if k = 0, the claim is true: 20 = 1, and we only have one node (root) at level 0. (inductive step): suppose the claim is true for k-1: level k-1 contains 2k-1 nodes. Since each node at level k-1 has 2 children, there are twice as many nodes at level k. So, level k contains 2 * 2k-1 = 2k nodes. © 2004 Goodrich, Tamassia Trees How many nodes in a tree of height h? What is the height of a tree of size n (with n nodes)? Theorem: A proper binary tree of height h contains We know that n = 2h+1 – 1. 2 h+1 - 1 nodes. So, 2h+1 = n + 1. Proof: by induction on h • • h + 1 = log 2 (n+1) (basis of induction): h=0. The tree contains 21 - 1 = 1 node. h = log 2 (n+1) – 1. (inductive step): assume a tree of height h-1 contains 2h - 1 nodes. A tree of depth h has one more level (h) which contains 2h nodes. The total number of nodes in the tree of height h is: 2 h - 1 + 2 h = 2 * 2 h – 1 = 2 h+1 - 1. © 2004 Goodrich, Tamassia 20 Trees 21 So, the height of the tree is logarithmic in the size of the tree. The size of the tree is exponential in the height (number of levels) of the tree. © 2004 Goodrich, Tamassia Trees 22 4