Disusun Oleh : Budi Arifitama Pertemuan ke-8

Define trees as data structures Define the terms associated with trees Discuss tree traversal algorithms Discuss Binary Tree Examine a binary tree example

 Tree Introduction  Term Associated With Tree  Binary Tree  Traverse Tree

A tree is a nonlinear data structure used to represent entities that are in some hierarchical relationship Examples in real life: Family tree Table of contents of a book Class inheritance hierarchy in Java Computer file system (folders and subfolders) Decision trees Top-down design 10-4

10-5 Root directory of C drive Documents and SettingsProgram FilesMy Music DesktopFavoritesStart MenuMicrosoft OfficeAdobe

Tree: a set of elements of the same type such that It is empty Or, it has a distinguished element called the root from which descend zero or more trees (subtrees) What kind of definition is this? What is the base case? What is the recursive part? 10-7

10-8 Leaf nodes Root Interior nodes

Nodes: the elements in the tree Edges: connections between nodes Root: the distinguished element that is the origin of the tree There is only one root node in a tree Leaf node: a node without an edge to another node Interior node: a node that is not a leaf node Empty tree has no nodes and no edges 10-9

Parent or predecessor: the node directly above in the hierarchy A node can have only one parent Child or successor: a node directly below in the hierarchy Siblings: nodes that have the same parent Ancestors of a node: its parent, the parent of its parent, etc. Descendants of a node: its children, the children of its children, etc Tree Terminology

Does a leaf node have any children? Does the root node have a parent? How many parents does every node other than the root node have? 10-11

A path is a sequence of edges leading from one node to another Length of a path: number of edges on the path Height of a (non-empty) tree : length of the longest path from the root to a leaf What is the height of a tree that has only a root node? By convention, the height of an empty tree is

Level of a node : number of edges between root and node It can be defined recursively: Level of root node is 0 Level of a node that is not the root node is level of its parent + 1 Question: What is the level of a node in terms of path length? Question: What is the height of a tree in terms of levels? 10-13

10-14 Level 0 Level 1 Level 2 Level 3

Subtree of a node: consists of a child node and all its descendants A subtree is itself a tree A node may have many subtrees 10-15

10-16 Subtrees of the root node

10-17 Subtrees of the node labeled E E

Degree or arity of a node: the number of children it has Degree or arity of a tree: the maximum of the degrees of the tree’s nodes 10-18

Ancestor (F)? Predesesor (F) ? Descendant (B)? Succesor (B)? Parent (I)? Child (C)? Sibling (G)? Size? Height? Root? Leaf? Degree (C)?

Gambarkan tree dari representasi berikut : D  F D  B K  J K  L B  A B  C H  D H  K F  E F  G J  I Tentukan : 1.Root 2.Leaf 3.Heigth 4.Child H 5.Parent A

 Finite (possibly empty) collection of elements.  A nonempty binary tree has a root element.  The remaining elements (if any) are partitioned into two binary trees.  These are called the left and right subtrees of the binary tree.

23  There are many variations on trees but we will start with binary trees  binary tree: each node has at most two children  the possible children are usually referred to as the left child and the right child parent left child right child

24  full binary tree: a binary tree is which each node was exactly 2 or 0 children

 What is the maximum height of a full binary tree with 11 nodes? A. 1 B. 3 C. 5 D. 7 E. Not possible to construct a full binary tree with 11 nodes. CS314 Binary Trees 25

26  complete binary tree: a binary tree in which every level, except possibly the deepest is completely filled. At depth n, the height of the tree, all nodes are as far left as possible Where would the next node go to maintain a complete tree? Binary Trees CS314

27  perfect binary tree: a binary tree with all leaf nodes at the same depth. All internal nodes have exactly two children.  a perfect binary tree has the maximum number of nodes for a given height  a perfect binary tree has (2 (n+1) - 1) nodes where n is the height of the tree  height = 0 -> 1 node  height = 1 -> 3 nodes  height = 2 -> 7 nodes  height = 3 -> 15 nodes Binary Trees CS314

 No node in a binary tree may have a degree more than 2, whereas there is no limit on the degree of a node in a tree.  A binary tree may be empty; a tree cannot be empty.

 The subtrees of a binary tree are ordered; those of a tree are not ordered. a b a b Are different when viewed as binary trees. Are the same when viewed as trees.

 Representasi ekspresi arithmatik

 Minimum number of nodes in a binary tree whose height is h.  At least one node at each of first h levels. minimum number of nodes is h

 All possible nodes at first h levels are present. Maximum number of nodes = … + 2 h-1 = 2 h - 1

 A binary tree can bee represented by an Array or a linked list.

HDKBFJLACEGI Array

H KD B A L J I leftChild elementrightChild root F C E G

tree[] 510 abcdefghij b a c d e f g hi j

a b 1 3 c 7 d 15

a b 1 3 c 7 d tree[] 0510 a-b---c d

 Penelusuran seluruh node pada binary tree.  Metode :  Preorder  Inorder  Postorder  Level order

 Preorder traversal 1. Cetak data pada root 2. Secara rekursif mencetak seluruh data pada subpohon kiri 3. Secara rekursif mencetak seluruh data pada subpohon kanan

a bc a b c

a bc d e f g hi j abdgheicfj

+ ab - c d + ef * / Gives prefix form of expression! /*+ab-cd+ef

 Inorder traversal 1. Secara rekursif mencetak seluruh data pada subpohon kiri 2. Cetak data pada root 3. Secara rekursif mencetak seluruh data pada subpohon kanan

a bc bac

a bc d e f g hi j gdhbeiafjc

 Postorder traversal 1. Secara rekursif mencetak seluruh data pada subpohon kiri 2. Secara rekursif mencetak seluruh data pada subpohon kanan 3. Cetak data pada root

a bc bca

a bc d e f g hi j ghdiebjfca

+ ab - cd + e f * / Gives postfix form of expression! ab+cd-*ef+/

 Telusuri pohon biner berikut dengan menggunakan metode pre, in, post, dan level traversal.

a.b.