Binary Trees 2010 Goodrich Tamassia Trees 1 Binary

Binary Tree (BT) q A binary tree is a tree with the following properties: n n q q q n Each internal node has one or two children (exactly two for proper binary trees) The children of a node are an ordered pair n n n 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 arithmetic expressions decision processes searching A We call the children of an internal node left child and right child Alternative recursive definition: a binary tree is either n Applications: B C D E F G H © 2010 Goodrich, Tamassia Trees 2

Example: Arithmetic Expression Trees q Binary tree associated with an arithmetic expression n n q internal nodes: operators external nodes: operands Example: arithmetic expression tree for the expression (2 (a - 1) + (3 b)) Proper binary tree if no unary operator! + - 2 a © 2010 Goodrich, Tamassia 3 b 1 Trees 3

Example: Decision Trees q Binary tree associated with a decision process n n q internal nodes: questions with yes/no answer external nodes: decisions Example: dining decision Proper binary tree! Want a fast meal? No Yes How about coffee? On expense account? Yes No Starbucks Spike’s Al Forno Café Paragon © 2010 Goodrich, Tamassia Trees 4

Properties of Binary Trees q Notation Properties: n 1 ne 2 h n h ni 2 h-1 n h+1 n 2 h+1 -1 n log 2(n+1)-1 h n-1 n: # of nodes ne: # of external nodes ni: # of internal nodes h: height Quiz! level=0 Level d has at most 2 d nodes. level=1 level=2 level=3 © 2010 Goodrich, Tamassia Trees 6

Properties of Proper Binary Trees q Notation Quiz! Properties: n h+1 ne 2 h n h ni 2 h-1 n 2 h+1 n 2 h+1 -1 n log 2(n+1)-1 h (n-1)/2 n ne = ni+1 n: # of nodes ne: # of external nodes ni: # of internal nodes h: height level=0 Proper! Level d has at most 2 d nodes. level=1 level=2 level=3 © 2010 Goodrich, Tamassia Trees 7

Proof by Induction: ne = ni+1 for Proper Binary Trees Prove that ne = ni+1 n When ni = 1. . . n When ni = 2. . . n Assume the identity holds when ni k, then when ni = k+1 … ni = 1 ni = 2 © 2010 Goodrich, Tamassia ni = k+1 Trees 8

Quiz Given a proper binary tree with n node n Explain why n is

Proof by Induction: n 0 = n 2+1 for General Binary Trees nk :

Binary. Tree ADT q q The Binary. Tree ADT extends the Tree ADT, i. e. , it inherits all the methods of the Tree ADT Additional methods: n n position p. left() position p. right() © 2010 Goodrich, Tamassia Trees q q Update methods may be defined by data structures implementing the Binary. Tree ADT Proper binary tree: Each node has either 0 or 2 children 12

Linked Structure for Trees q A node is represented by an object storing n n n Element Parent node Sequence of children nodes B Link to list of pointers to children A D F B D A C © 2010 Goodrich, Tamassia F Link to parent C E Trees E 13

Linked Structure for Binary Trees q A node is represented by an object storing

Vector Representation of Binary Trees q Nodes of tree T are stored in vector S 1 0 A B D 1 2 3 … G H 10 11 A … 2 B q Node n n n v is stored at S[f(v)] f() is known as level numbering f(root) = 1 4 if v is the left child of parent(v), E f(v) = 2*f(parent(v)) if v is the right child of parent(v), f(v) = 2*f(parent(v)) + 1 Quiz! © 2010 Goodrich, Tamassia Trees 3 D 5 6 7 C F 10 J 11 G H 15

Vector Representation of Binary Trees: More Examples © 2010 Goodrich, Tamassia Trees 16

Quiz q 1 The binary tree is stored level-by-level in a one-dimensional array. What

Vector Representation of Binary Trees: Analysis q Notation n n q Time complexity q Space complexity n: # of nodes in tree T N: size of vector S n © 2010 Goodrich, Tamassia Trees O(N), which is O(2 n) in the worse case Major drawback! (So we always want to keep trees as shallow as possible!) 18

Traversal of Binary Trees q Basic types of traversal n n Preorder Postorder Inorder

Postorder Traversal for BT q Can be applied to any “bottom-up” evaluation problems n n Evaluate an arithmetic expression Directory size computation (for general trees) © 2010 Goodrich, Tamassia Trees 21

Evaluate Arithmetic Expressions q Based on postorder traversal n n Recursive method returning the value of a subtree When visiting an internal node, combine the values of the subtrees + - 2 5 © 2010 Goodrich, Tamassia 3 Algorithm eval. Expr(v) if v. is. External() return v. element() x eval. Expr(v. left()) y eval. Expr(v. right()) à operator stored at v return x y Postorder traversal ≡ Postfix notation 2 1 Trees 22

Inorder Traversal q q Inorder traversal: a node is visited after its left subtree and before its right subtree Application n n Algorithm in. Order(v) if v is NULL return in. Order(v. left()) visit(v) in. Order(v. right()) Draw a binary tree Print arithmetic expressions with parentheses 6 2 8 1 4 3 © 2010 Goodrich, Tamassia 7 Inorder traversal ≡ Projection! 9 5 Trees 23

Inorder Traversal: Examples q Properties of inorder traversal n n Very close to infix

Print Arithmetic Expressions q Based on inorder traversal n n n Print operand or operator when visiting node Print “(“ before traversing left subtree Print “)“ after traversing right subtree + - 2 a 3 Algorithm print. Expression(v) if v is NULL return print(“(’’) in. Order(v. left()) print(v. element()) in. Order(v. right()) print (“)’’) b ((2 (a - 1)) + (3 b)) 1 © 2010 Goodrich, Tamassia Trees 25

Draw a Binary Tree q Since inorder traversal is equivalent to tree projection, it

Euler Tour Traversal q q q 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: n On the left (preorder) + x 2 – 5 1 x 3 2 n From below (inorder) 2 x 5 – 1 + 3 x 2 n On the right (postorder) 2 5 1 – x 3 2 x + + Quiz! L 2 R B 5 © 2010 Goodrich, Tamassia 3 2 1 Trees 27

Euler Tour Traversal q Applications n n Determine the number of descendants of each node in a tree Fully parenthesize an arithmetic expression from an expression tree © 2010 Goodrich, Tamassia Trees 28

Quiz q Determine a binary tree n n q Inorder=[c q f p x z k y] Preorder=[f c q z x p y k] Inorder=[z b j d e f m c] Postorder=[b z d e m c f j] Determine a binary tree n n Preorder=[a b c] Postorder=[c b a] © 2010 Goodrich, Tamassia Trees 29