Trees1 Make Money Fast! Stock Fraud Ponzi Scheme Bank Robbery © 2010 Goodrich, Tamassia

Trees2 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 SalesR&DManufacturing LaptopsDesktops US International EuropeAsiaCanada © 2010 Goodrich, Tamassia

Trees3 subtree Tree 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. A B DC GH E F IJ K  Subtree: tree consisting of a node and its descendants © 2010 Goodrich, Tamassia

Examples of Trees  File system Internal nodes: directories (folders) External nodes: files © 2010 Goodrich, TamassiaTrees4

5 Tree ADT  We use positions to abstract nodes  Generic methods: integer size() boolean empty()  Accessor methods: position root() list positions()  Position-based methods: position p.parent() list p.children() Query methods: boolean p.isRoot() boolean p.isExternal() Additional update methods may be defined for specific applications © 2010 Goodrich, Tamassia position = pointer list = linked list or vector

Trees6 Preorder Traversal  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. MotivationsReferences2. Methods 2.1 Stock Fraud 2.2 Ponzi Scheme 1.1 Greed1.2 Avidity 2.3 Bank Robbery Algorithm preOrder(v) visit(v) for each child w of v preorder (w) © 2010 Goodrich, Tamassia

Preorder Traversal: Example © 2010 Goodrich, TamassiaTrees7

8 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) cs16/ homeworks/ todo.txt 1K programs/ DDR.cpp 10K Stocks.cpp 25K h1c.doc 3K h1nc.doc 2K Robot.cpp 20K © 2010 Goodrich, Tamassia

Postorder Traversal: Example © 2010 Goodrich, TamassiaTrees9

Postorder Traversal: Example 2  We want to use postorder traversal to calculate the disk usage of a folder “cs016”. © 2010 Goodrich, TamassiaTrees10

Iterative Implementation of Preorder Traversal  Steps 1. Push the root to the stack 2. Pop the stack and visit it 3. Push the children in a reverse order 4. Repeat 2) and 3) until the stack is empty © 2010 Goodrich, TamassiaTrees17

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees18  A B DC GH E F IJ K A Preorder: Stack

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees19  A B DC GH E F IJ K BCDBCD Preorder: A

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees20  A B DC GH E F IJ K EFCDEFCD Preorder: A B

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees21  A B DC GH E F IJ K FCDFCD Preorder: A B E

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees22  A B DC GH E F IJ K IJKCDIJKCD Preorder: A B E F

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees23  A B DC GH E F IJ K JKCDJKCD Preorder: A B E F I

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees24  A B DC GH E F IJ K KCDKCD Preorder: A B E F I J

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees25  A B DC GH E F IJ K CDCD Preorder: A B E F I J k

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees26  A B DC GH E F IJ K GHDGHD Preorder: A B E F I J k C

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees27  A B DC GH E F IJ K HDHD Preorder: A B E F I J k C G

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees28  A B DC GH E F IJ K D Preorder: A B E F I J k C G H

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees29  A B DC GH E F IJ K Preorder: A B E F I J k C G H D

Iterative Implementation of Preorder Traversal: Demo © 2010 Goodrich, TamassiaTrees30  A B DC GH E F IJ K Preorder: A B E F I J k C G H D Done!

Iterative Implementation of Postorder Traversal  Steps 1. Push the root node to stack1. 2. Pop a node from stack 1, and push it to stack Then push its children sequentially to stack Repeat step 2) and 3) until stack 1 is empty. 5. Pop all nodes from stack 2 to obtain the traversal in postorder. © 2010 Goodrich, TamassiaTrees31

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees32  A B DC GH E F IJ K A Stack 1Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees33  A B DC GH E F IJ K DCBDCB Stack 1 A Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees34  A B DC GH E F IJ K CBCB Stack 1 DADA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees35  A B DC GH E F IJ K HGBHGB Stack 1 CDACDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees36  A B DC GH E F IJ K GBGB Stack 1 HCDAHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees37  A B DC GH E F IJ K B Stack 1 GHCDAGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees38  A B DC GH E F IJ K FEFE Stack 1 BGHCDABGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees39  A B DC GH E F IJ K KJIEKJIE Stack 1 FBGHCDAFBGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees40  A B DC GH E F IJ K JIEJIE Stack 1 KFBGHCDAKFBGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees41  A B DC GH E F IJ K IEIE Stack 1 JKFBGHCDAJKFBGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees42  A B DC GH E F IJ K E Stack 1 IJKFBGHCDAIJKFBGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees43  A B DC GH E F IJ K Stack 1 EIJKFBGHCDAEIJKFBGHCDA Stack 2

Iterative Implementation of Postorder Traversal: Demo © 2010 Goodrich, TamassiaTrees44  A B DC GH E F IJ K Stack 1 EIJKFBGHCDAEIJKFBGHCDA Stack 2 Postorder: E I J K F B G H C D A By popping all elements in stack 2

Other Traversal  Breadth-first traversal (level-order traversal) Idea: Visit all the nodes at depth d before visiting the nodes at depth d+1 Implementation: Using a queue © 2010 Goodrich, TamassiaTrees45 A B DC GH E F IJ K level-order traversal: A B C D E F G H I J K

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees46 A A B DC GH E F IJ K Output: End Queue Front

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees47 BCD A B DC GH E F IJ K Output: A Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees48 CDEF A B DC GH E F IJ K Output: AB Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees49 DEFGH A B DC GH E F IJ K Output: ABC Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees50 EFGH A B DC GH E F IJ K Output: ABCD Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees51 FGH A B DC GH E F IJ K Output: ABCDE Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees52 GHIJK A B DC GH E F IJ K Output: ABCDEF Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees53 HIJK A B DC GH E F IJ K Output: ABCDEFG Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees54 IJK A B DC GH E F IJ K Output: ABCDEFGH Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees55 JK A B DC GH E F IJ K Output: ABCDEFGHI Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees56 K A B DC GH E F IJ K Output: ABCDEFGHIJ Queue

Breadth-first Traversal: Demo © 2010 Goodrich, TamassiaTrees57 A B DC GH E F IJ K Output: ABCDEFGHIJK Queue Breadth-first traversal!

Trees58 Binary Trees  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 a binary tree  Applications: arithmetic expressions decision processes searching A B C FG D E H I © 2010 Goodrich, Tamassia

Trees59 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 a1 3b © 2010 Goodrich, Tamassia

Trees60 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? StarbucksSpike’sAl FornoCafé Paragon Yes No YesNoYesNo © 2010 Goodrich, Tamassia

Trees61 Properties of Binary Trees  Notation n : # of nodes n e : # of external nodes n i : number of internal nodes h : height Properties: h  1  n  2 h+1  1 1  n e  2 h h  n i  2 h  1 log 2 (n  1)-1  h  n  1 © 2010 Goodrich, Tamassia level=0 level=1 level=2 level=3 Level d has at most 2 d nodes.

Trees62 Properties of Proper Binary Trees  Notation n : # of nodes n e : # of external nodes n i : number of internal nodes h : height Properties: 2h  1  n  2 h+1  1 h  1  n e  2 h h  n i  2 h  1 log 2 (n  1)-1  h  n  1)/2 n e  n i  1 © 2010 Goodrich, Tamassia level=0 level=1 level=2 level=3 Level d has at most 2 d nodes. Proper!

Trees63 BinaryTree ADT  The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT  Additional methods: position p.left() position p.right()  Update methods may be defined by data structures implementing the BinaryTree ADT  Proper binary tree: Each node has either 0 or 2 children © 2010 Goodrich, Tamassia

Trees64  Linked Structure for Trees  A node is represented by an object storing Element Parent node Sequence of children nodes  Node objects implement the Position ADT B D A CE F B  ADF  C  E © 2010 Goodrich, Tamassia

Trees65 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 D A CE   B AD CE  © 2010 Goodrich, Tamassia

Vector Representation of Binary Trees  Nodes of a tree T are stored in an vector S © 2010 Goodrich, Tamassia66Trees  Node v is stored at S[f(v)] f(root) = 1 if v is the left child of parent(v), f(v) = 2*f(parent(v)) if v is the right child of parent(v), f(v) = 2*f(parent(v))  A HG FE D C B J ABDGH … … f() is known as level numbering

Vector Representation of Binary Trees: More Examples © 2010 Goodrich, Tamassia67Trees

Vector Representation of Binary Trees: Analysis  Notation n: # of nodes in T N: size of S  Time complexity  Space complexity O(N), which is O(2 n ) in the worse case © 2010 Goodrich, TamassiaTrees68 Major weakness! (So we always want to keep trees as shallow as possible!)

Traversal of Binary Trees  TypesInorder traversal: a node is visited after its left subtree and before its right subtree  Application: draw a binary tree x(v) = inorder rank of v y(v) = depth of v © 2010 Goodrich, TamassiaTrees

Traversal of Binary Trees  Basic types of traversal Preorder Postorder Inorder Level order © 2010 Goodrich, TamassiaTrees70

Postorder Traversal for BT  Can be applied to any “bottom-up” evaluation problems Evaluate an arithmetic expression Directory size computation © 2010 Goodrich, TamassiaTrees71

Trees72 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 v.isExternal() return v.element() else x  evalExpr(v.left()) y  evalExpr(v.right())   operator stored at v return x  y    © 2010 Goodrich, Tamassia Postorder traversal ≡ Postfix notation

Trees73 Inorder Traversal  Inorder traversal: a node is visited after its left subtree and before its right subtree  Application Draw a binary tree Print arithmetic expressions with parentheses Algorithm inOrder(v) if  v.isExternal() inOrder(v.left()) visit(v) if  v.isExternal() inOrder(v.right()) © 2010 Goodrich, Tamassia Inorder traversal ≡ Projection!

Inorder Traversal: Examples  Properties of inorder traversal Very close to infix notation Can be obtained by tree projection © 2010 Goodrich, TamassiaTrees74

Trees75 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 Algorithm printExpression(v) if  v.isExternal() print( “(’’ ) inOrder(v.left()) print(v.element()) if  v.isExternal() inOrder(v.right()) print ( “)’’ )    2 a1 3b ((2  ( a  1))  (3  b)) © 2010 Goodrich, Tamassia

Draw a Binary Tree © 2010 Goodrich, TamassiaTrees76  Since inorder traversal is equivalent to tree projection, it is easy to draw a binary tree using inorder traversal.

Trees77 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)  + x 2 – 5 1 x 3 2 from below (inorder)  2 x 5 – x 2 on the right (postorder)  – x 3 2 x +    L B R  © 2010 Goodrich, Tamassia

Euler Tour Traversal  Applications Determine the number of descendants of each node in a tree Fully parenthesize an arithmetic expression from an expression tree © 2010 Goodrich, TamassiaTrees78

From General to Binary Trees  How to transform a general tree to a binary tree  Left-child right-sibling representation © 2010 Goodrich, TamassiaTrees79