Trees 4/23/2017 Trees

What is a Tree 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 Computers”R”Us Sales R&D Manufacturing Laptops Desktops CAN International Europe Asia USA Last Update: Oct 16, 2014 Trees

Tree Terminology For a node x: Ancestors: x, parent, grandparent, great-grandparent, etc. Descendants: x, children, grandchildren, great-grandchildren, etc. Proper ancestors/descendants: exclude x itself. Depth: number of proper ancestors of x. For a tree: Root: node without parent (A) Internal node: node with at least one child (A, B, C, F) External node (aka, leaf ): node without children (E, I, J, K, G, H, D) Height: maximum node depth (3) Subtree: tree consisting of a node and its descendants A B C D E F G H I J K Last Update: Oct 16, 2014 Trees

A recursive view of Trees Tk Last Update: Oct 16, 2014 Trees

Tree ADT Query methods: We use positions to abstract nodes Generic methods: integer size() boolean isEmpty() Iterator iterator() Iterable positions() Accessor methods: position root() position parent(p) Iterable children(p) Integer numChildren(p) Query methods: boolean isInternal(p) boolean isExternal(p) boolean isRoot(p) Additional update methods may be defined by data structures implementing the Tree ADT Last Update: Oct 16, 2014 Trees

Java Tree interface Last Update: Oct 16, 2014 Trees

Tree Traversals A traversal method is a systematic way to explore the tree structure by visiting its nodes in a specific order. Very useful computational tool with many applications. Important tree traversals: Preorder Postorder Inorder Euler tour (generalizes the above three) Level order (aka, Breadth First Search)  not shown in these slides. Generalized BFS on graphs will be discussed later in the course. Last Update: Oct 16, 2014 Trees

Preorder Traversal Algorithm preOrder(v) visit(v) for each child 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 Make Money Fast! 1. Motivations References 2. Methods 2.1 Stock Fraud 2.2 Ponzi Scheme 1.1 Greed 1.2 Avidity 2.3 Bank Robbery 1 2 3 5 4 6 7 8 9 Last Update: Oct 16, 2014 Trees

Postorder Traversal In a postorder traversal, a node is visited after its descendants Application: compute space used by files in a directory and its subdirectories Algorithm postOrder(v) for each child w of v postOrder (w) visit(v) 9 EECS2011/ 8 3 7 todo.txt 1K assignments/ programs/ 1 2 4 5 6 A1.doc 3K A2.doc 2K LinOpt.java 10K Stocks.java 25K Robot.java 20K Last Update: Oct 16, 2014 Trees

Binary Trees Applications: A binary tree is a tree such that: 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 A leaf node has no children Recursive definition of a binary tree T: T consists of an external root node, or T has internal root whose left and right subtrees are binary trees. Applications: arithmetic expressions decision processes searching A B C F G D E H I J Last Update: Oct 16, 2014 Trees

A recursive view of Binary Trees left subtree of node right subtree of Last Update: Oct 16, 2014 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) ) +  - 2 a 1 3 b Last Update: Oct 16, 2014 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? How about coffee? On expense account? Starbucks Chipotle Gracie’s Café Paragon Yes No Last Update: Oct 16, 2014 Trees

Properties of Proper Binary Trees Notation: n number of nodes e number of external nodes i number of internal nodes h height Properties: e = i + 1 n = i + e = 2e - 1 h  i h  (n - 1)/2 e  2h h  log2 e h  log2 (n + 1) - 1 Last Update: Oct 16, 2014 Trees

BinaryTree ADT The BinaryTree ADT extends the Tree ADT, i.e., inherits all methods of Tree ADT Additional methods: position left(p) position right(p) position sibling(p) boolean isInternal(p) boolean isExternal(p) The above position methods return null when there is no left, right, or sibling of p, respectively Update methods may be defined by data structures implementing the BinaryTree ADT Last Update: Oct 16, 2014 Trees

Inorder Traversal Algorithm inOrder(v) if isInternal(v) then inOrder( left(v) ) visit(v) if isInternal(v) then inOrder( right(v) ) In an inorder traversal a node is visited after its left subtree and before its right subtree Application: draw a binary tree. Planar node coordinates: x(v) = inorder rank of v y(v) = depth of v 3 1 2 5 6 7 9 8 4 Last Update: Oct 16, 2014 Trees

Print Arithmetic Expressions Algorithm printExpr(v) if isInternal(v) then print( “(” ) printExpr ( left(v) ) print( v.element () ) if isInternal(v) then printExpr ( right(v) ) print ( “)” ) Specialization of inorder traversal print operand or operator when visiting node print “(“ before traversing left subtree print “)“ after traversing right subtree +  - 2 a 1 3 b ((2  (a - 1)) + (3  b)) Last Update: Oct 16, 2014 Trees

Evaluate Arithmetic Expressions Specialization of postorder: recursive method returning the value of a subtree when visiting an internal node, combine the values of its left & right subtrees Algorithm evalExpr(v) if isExternal (v) then return v.element() else x  evalExpr(left(v)) y  evalExpr(right(v))   v.element() return x  y +  - 2 5 1 3 Last Update: Oct 16, 2014 Trees

Euler Tour Traversal Generic traversal of a binary tree Includes as special cases: preorder, postorder and inorder Walk around the tree and visit each node three times: on the left (preorder) from below (inorder) on the right (postorder) +  - 2 5 1 3 L B R Last Update: Oct 16, 2014 Trees

Template Method Pattern public abstract class EulerTour { protected BinaryTree tree; protected void visitExternal(Position p, Result r) { } protected void visitLeft(Position p, Result r) { } protected void visitBelow(Position p, Result r) { } protected void visitRight(Position p, Result r) { } protected Object eulerTour(Position p) { Result r = new Result(); // local variable 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 can be redefined by subclasses Template method eulerTour Recursively called on the left and right children A local variable r of type Result with fields leftResult, rightResult and finalResult keeps track of the output of the recursive calls to eulerTour Last Update: Oct 16, 2014 Trees

Specializations of EulerTour public class EvaluateExpression extends EulerTour { protected void visitExternal(Position p, Result r) { r.finalResult = (Integer) p.element(); } protected void visitRight(Position p, Result r) { Operator op = (Operator) p.element(); r.finalResult = op.operation( (Integer) r.leftResult, (Integer) r.rightResult ); } // … the rest omitted … } 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) Last Update: Oct 16, 2014 Trees

Linked Structure for Trees  B A D F C E element parent children child list node A node is represented by an object storing Element Parent node list of children nodes Node objects implement the Position ADT B D A C E F Last Update: Oct 16, 2014 Trees

Linked Structure for 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 B D A C E Last Update: Oct 16, 2014 Trees

Array-Based Representation of Binary Trees 1 2 5 6 3 4 9 10 a h g f e d c b j Nodes are stored in an array A a b d g h … 1 2 9 10 Node v is stored at A[rank(v)] rank(root) = 0 rank(left(node)) = 2  rank(node) + 1 rank(right(node)) = 2  rank(node) + 2 rank(parent(node)) = (rank(node) -1)/2 Last Update: Oct 16, 2014 Trees

Comparison Linked Structure Array Requires explicit representation of 3 links per position: parent, left child, right child Data structure grows as needed – no wasted space. Parent and children are implicitly represented: Lower memory requirements per position Memory requirements determined by height of tree. If tree is sparse, this is highly inefficient. Last Update: Oct 16, 2014 Trees

Summary The Tree ADT, tree terminologies, and Java interface Tree Traversals Preorder Postorder Inorder Euler Tour Level order (aka, breadth first search) Binary trees: properties & some applications Linked list & array based representations of trees. Last Update: Oct 16, 2014 Trees

Last Update: Oct 16, 2014 Trees