Data Structure - Binary Tree 1 -

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1 Data Structure - Binary Tree 1 - Hanyang University Jong-Il Park

2 Basic Tree Concepts Logical structures Chap. 2~4 Chap. 5 Chap. 6 Linear list Tree Graph Linear structures Non-linear structures

3 Linear Lists and Trees Linear lists are useful for serially ordered data. (e 0, e 1, e 2,, e n-1 ) Days of week Months in a year Students in this class Trees are useful for hierarchically ordered data. Employees of a corporation. President, vice presidents, managers, and so on.

4 Hierarchical Data and Trees The element at the top of the hierarchy is the root. Elements next in the hierarchy are the children of the root. Elements next in the hierarchy are the grandchildren of the root, and so on. Elements that have no children are leaves.

5 Example Tree President root children of root VP1 VP2 VP3 Manager1 Manager2 Manager Manager grand children of root Worker Bee great grand child of root

6 Example Tree two types of genealogical charts

Terminology(Cont.) Def. Tree : A finite set of one or more nodes s.t. (1) There is a specially designated node called the root (2) The remaining nodes are partitioned into n disjoint sets T 1,.

7 Terminology(Cont.) Def. Tree : A finite set of one or more nodes s.t. (1) There is a specially designated node called the root (2) The remaining nodes are partitioned into n disjoint sets T 1,.., T n, where each of these sets is a tree. T 1,.., T n are called the subtrees of the root. node, branch degree # of subtrees of a node

8 Terminology (Cont.) leaf or terminal node nodes that have degree zero nonterminal nodes siblings children of the same parent degree of a tree the maximum of the degree of the nodes in the tree ancestors of a node all the nodes along the path from the root to that node

9 Terminology (Cont.) level of a node defined by letting the root be at level one height or depth of a tree the maximum level of any node in the tree

10 Representation of trees: 1) Left child-right sibling representation every node has at most one leftmost child and at most one closest right sibling

11 Representation of trees: 2) Representation as a degree-two tree have the two children of a node binary trees left child-right child trees

12 Representation of trees: 2) Representation as a degree-two tree (Cont.)

13 BINARY TREES

14 Binary trees Definition: A Tree in which each node has no more than 2 children (left subtree and right subtree) Typical Representation in C typedef struct Tree_Node *PtrToNode; typedef struct PtrToNode Tree; r struct Tree_Node { ElementType Element; Tree Left; Tree Right; }; T L T R Division of 16Computer Science and Engineering, Hanyang University

15 The abstract data type a binary tree empty tree having zero nodes consist of a root and two disjoint binary trees (left subtree and right subtree) Binary tree Binary search tree

16 The abstract data type(cont.) distinctions between a binary tree and a tree a binary tree is a set of nodes either empty or there is no tree having zero nodes but there is an empty binary tree in a binary tree, distinguish between the order of the children; in a tree we do not the same as a tree

17 The abstract data type (Cont.)

18 The abstract data type (Cont.) a skewed binary tree: Figure 5.10(a) a complete binary tree: Figure 5.10(b)

19 Properties of binary trees Lemma 5.2 [Maximum number of nodes] 1) maximum number of nodes on level i of a binary tree: 2 i-1 2) maximum number of nodes in a binary tree of depth k : 2 k 1 Sum of max. no. of nodes on level i Lemma 5.3 [Relation between # of leaf nodes and degree-2 nodes] n 0 : # of leaf nodes n 2 : # of nodes of degree 2 n 0 = n 2 + 1

20 Properties of binary trees (Cont.) Lemma 5.3 [Relation between # of leaf nodes and degree-2 nodes] (Cont.) Proof: n 1 : # of nodes of degree one n : the total number of nodes n = n 0 + n 1 + n (5.1) B : the number of branches n = B + 1 every node except the root has a branch leading into it B = n 1 + 2n 2 n = B + 1 = n 1 + 2n (5.2) n 0 = n from Eq. 5.1 and 5.2

21 Properties of binary trees (Cont.) Definition: a full binary tree of depth k is a binary tree of depth k having 2 k - 1 nodes Definition: a binary tree with n nodes and depth k is complete iff its nodes correspond to the nodes numbered from 1 to n in the full binary tree of depth k

22 BINARY TREE REPRESENTATION

23 Array representation can easily determine the locations of the parent, left child, and right child of any node Lemma 5.4: in a complete binary tree, for any node i 1) parent(i) = i /2 2) leftchild(i) = 2i 3) rightchild(i) = 2i + 1

24 Array representation (Cont.) for the skewed tree, space is wasted

25 Linked representation general inadequacies of sequential representation wasteful for a skewed tree insertion and deletion of nodes from the middle of a tree require memory movement linked representation typedef struct node *treepointer; typedef struct node { int data; treepointer leftchild, rightchild; };

26 Linked representation (Cont.) /* global variable */ treepointer root;

27 Application of binary trees Binary Search Tree: representing ordered sets of elements Expression Tree: intermediate representation for expressions used by the compiler Expression Tree: An Example (a+b*c)+((d*e+f)*g) + + * a * + g b c * f d e Division 29 of Computer Science and Engineering, Hanyang University

28 BINARY TREE TRAVERSAL

29 Introduction traverse a tree or visit each node in the tree exactly once the order in which the nodes are visited let L, V, and R stand for moving left, visiting the node, and moving right six traversals: LVR, LRV, VLR, VRL, RVL, RLV consider only that we traverse left before right LVR: inorder -> infix LRV: postorder -> postfix VLR: preorder -> prefix

30 Traversal of binary trees Three ways to visit or traverse every node of a tree T, whose root is r and whose subtrees are T 1, T 2,..., T k. Preorder: Visit r, then recursively do a pre-order traversal of T 1, T 2,..., T k. [+,+,a,*,b,c,*,+,*,d,e,f,g] Example: Depth-First Search Postorder: Recursively do post-order traversal of T 1, T 2,..., T k, and then visit r. [a,b,c,*,+,d,e,*,f,+,g,*,+] Example: Code generation for Compilers Inorder: (for binary tree) Do in-order traversal of T L, visit r, then do inorder traversal of T R. [a,+,b,*,c,+,d,*,e,+,f,*,g] Example: Binary Search Tree Division 32 of Computer Science and Engineering, Hanyang University

31 Inorder traversal move left until we cannot go farther, and then visit the node, move one node to the right and continue

32 Inorder traversal (Cont.) - Left subtree root right subtree

33 Preorder traversal visit a node, traverse left, and continue if we cannot continue, move right and begin again or move back

34 Postorder traversal

35 Iterative inorder traversal to implement the tree traversal method without using recursion by using iterators

36 Level-Order Traversal a traversal that requires a queue visit the root first, then the root's left child, followed by the root's right child

37 Traversal without a stack 1. add a parent field to each node can trace way back up to any root and down again or 2. represent binary trees as threaded binary trees require two bits per node use the leftchild and rightchild fields to maintain the paths back to the root

38 ADDITIONAL BINARY TREE OPERATIONS

39 Copying binary trees Modification of the postorder traversal algorithm

40 Testing equality determine the equality of two binary trees equivalent if two binary trees have the same topology and the identical data topology: every branch in one tree corresponds to a branch in the second tree in the same order

41 The satisfiability problem the formulas of the propositional calculus a variable is an expression if x and y are expressions, x y, x y, x are expressions can use parentheses to alter the normal order of evaluation x 1 (x 2 x 3 ) the satisfiability problem for formulas of propositional calculus ask whether or not there is an assignment of values to the variables that causes the value of the expression to be true

42 The satisfiability problem (Cont.) (x 1 x 2 ) ( x 1 x 3 ) x 3 let (x 1, x 2, x 3 ) take on all possible combinations of true and false check the formula for each combination to evaluate an expression, need to traverse its tree in postorder

43 The satisfiability problem (Cont.) define this node structure in C True and False : constants typedef enum {not,and,or,true,false} logical; typedef struct node *treepointer; typedef struct node { treepointer leftchild; logical data; short int value; treepointer rightchild; };

44 The satisfiability problem (Cont.) See Program 5.9.

45 THREADED BINARY TREES

46 Threads there are more 0-links than actual pointers n links and 2n total links where # of nodes is n replace the 0-links by pointers, called threads 1) replace the rightchild field(value = 0) in node p by the inorder successor of p 2) replace the leftchild(value = 0) at node p by the inorder predecessor of p

47 Threads (Cont.) distinguish between threads and normal pointers add two boolean fields, leftthread and rightthread if t leftthread == TRUE, then t leftchild is a thread

48 Threads (Cont.) assume a head node for all threaded binary trees the original tree is the left subtree of the head node

49 Threads (Cont.)

50 Inorder traversal of a threaded binary tree if x rightthread == true, then the inorder successor of x is x rightchild if x rightthread == false, then the inorder successor of x is obtained by following a path of left-child links from the right child of x until a node with leftthread == true

51 Inorder traversal of a threaded binary tree (Cont.) Program 5.10

52 Inserting a node into a threaded binary tree how to make insertions into a threaded tree the case of inserting r as the right child of a node s if s has an empty right subtree, then a simple insertion if s has non-empty right subtree, then make this right subtree as the right subtree of r

53 Inserting a node into a threaded binary tree (Cont.)

54 Inserting a node into a threaded binary tree (Cont.)

Bioinformatics Programming. EE, NCKU Tien-Hao Chang (Darby Chang)

Bioinformatics Programming. EE, NCKU Tien-Hao Chang (Darby Chang) Bioinformatics Programming EE, NCKU Tien-Hao Chang (Darby Chang) 1 Tree 2 A Tree Structure A tree structure means that the data are organized so that items of information are related by branches 3 Definition