Data Structure. Chapter 5 Trees (Part II) Angela Chih-Wei Tang. National Central University Jhongli, Taiwan

Transcription

1 Data Structure Chapter 5 Trees (Part II) Angela Chih-Wei Tang Department of Communication Engineering National Central University Jhongli, Taiwan 2010 Spring

2 Threaded Binary Tree Problem: There are more null links than actual points How to make use of these null links? Threads! Threading rules If ptr->left_child = NULL ptr->left_child = inorder predecessor of ptr If ptr->right_child = NULL ptr->right_child = inorder successor of ptr C.E., NCU, Taiwan Angela Chih-Wei Tang,

3 Figure 5.21: Threaded Tree A B C D E F G H I Inorder sequence: H, D, I, B, E, A, F, C, G C.E., NCU, Taiwan Angela Chih-Wei Tang,

4 Threads (1/3) To distinguish between normal pointers and threads NEW!!! left_thread thread and right_thread thread typedef struct threaded_tree *threaded_pointer; typedef struct threaded_tree tree { short int left_thread; threaded_pointer left_child; char data; threaded_pointer right_child; Short int right_thread; }; LeftThread LeftChild RightChild RightThread data C.E., NCU, Taiwan Angela Chih-Wei Tang,

5 Threads (2/3) If ptr->left_thread = TRUE => ptr->left_child contains a thread If ptr->left_thread = FALSE => ptr->left_child contains a pointer to the left child If ptr->right_thread = TRUE ptr->right_child t i hild contains a thread If ptr->right_thread = FALSE => ptr->right_child contains a pointer to the right child C.E., NCU, Taiwan Angela Chih-Wei Tang,

6 Threads (3/3) To avoid dangling threads, a head node is used in representing a binary tree The original tree becomes the left subtree of the head node Empty Binary Tree LeftThread LeftChild TRUE data RightChild RightThread FALSE C.E., NCU, Taiwan Angela Chih-Wei Tang,

7 Memory Representation of Threaded Tree of Figure 5.20 f - f f A f f B f f B f f D f t E t f D f t E t t H t t I t C.E., NCU, Taiwan Angela Chih-Wei Tang,

8 Program 5.10: Finding the Inorder Successor of A Node threaded_pointer insucc(threaded_pointer tree) { /* find the inorder sucessor of tree in a threaded binary tree */ threaded_pointer temp; temp = tree->right_child; if(!tree->right_thread) while(!temp->left_thread) thread) temp = temp->left_child; return temp; } C.E., NCU, Taiwan Angela Chih-Wei Tang,

9 Fig. 5.24: Insertion of A Right Child in A Threaded Binary Tree (1/2) root root A A B parent B parent child C D C D child C.E., NCU, Taiwan Angela Chih-Wei Tang,

10 Fig. 5.24: Insertion of A Right Child in A Threaded Binary Tree (2/2) root root A A B B C D C X E F D X E F C.E., NCU, Taiwan Angela Chih-Wei Tang,

11 Program 5.12: Right Insertion in A Threaded Binary Tree void Insert_right(threaded_pointer parent, threaded_pointer child) { /* insert child as the right child of parent in a threaded binary tree */ threaded_pointer temp; child->right_child = parent->right_child; child->right_thread = parent->right_thread; child->left_child = parent; child->left_thread = TRUE; parent->right_child = child; parent->right_thread = FALSE; if (!child->right_thread) { temp = insucc(child); temp->left_child = child; } } C.E., NCU, Taiwan Angela Chih-Wei Tang,

12 Full Binary Tree A full binary tree of depth k is a binary tree of depth k having 2 k 1 nodes, k Depth=2 Depth=3 C.E., NCU, Taiwan Angela Chih-Wei Tang,

13 Complete Tree 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. Depth=

14 Binary Search Tree Binary search tree provides a better performance for search, insertion, and deletion. Definition: iti A binary search tree is a binary tree. If it is not empty, then, it satisfies the following properties: p Every element has a key and no two elements have the same key The keys (if any) in the left subtree are smaller than the key in the root. The keys (if any) in the right subtree are larger than the key in the root. The left and right subtrees are also binary search trees. C.E., NCU, Taiwan Angela Chih-Wei Tang,

15 Binary Search Trees Not binary search tree Binary search trees C.E., NCU, Taiwan Angela Chih-Wei Tang,

16 Program 5.15: Recursive Search of A Binary Search Tree tree_pointer search(tree_pointer root, int key) { /*return a pointer to the node that contains key. If there is no such node, return NULL. */ if (!root) return NULL; if (key == root->data) return root; if (key < root->data) return search(root->left_child, child key); return search(root->right_child, key); } C.E., NCU, Taiwan Angela Chih-Wei Tang,

17 Program 5.16: Iterative Search of A Binary Search Tree tree_pointer search2(tree_pointer tree, int key) { /* return a pointer to the node that contains key. If there is no such node, return NULL */ while (tree) { if (key == tree->data) return tree; if (key < tree->data) tree = tree->left_child; else tree = tree->right_child; } return NULL; } C.E., NCU, Taiwan Angela Chih-Wei Tang,

18 Program 5.17: Inserting An Element into A Binary Search Tree void insert_node(tree_pointer *node, int num) /* If num is in the tree pointed at by node do nothing; otherwise add a new node with data=num */ { /* modified_search: return the last node of the tree that was encountered during the search */ tree_pointer ptr, temp = modified_search(*node, num); if (temp!(*node)) { fprintf(stderr, The memory is full\n ); exit(1); } ptr->data = num; ptr->left_child = ptr->right_child = NULL; if (*node) /* insert as child of temp */ if (num < temp->data) temp->left_child=ptr; else temp->right_child = ptr; else *node = ptr; } C.E., NCU, Taiwan Angela Chih-Wei Tang,

19 Fig. 5.31: Inserting Into A Binary Search Tree Original tree Insert 80 Insert 35 C.E., NCU, Taiwan Angela Chih-Wei Tang,

20 Deletion From A Binary Search Tree Delete a leaf node A leaf node which is a right child of its parent A leaf node which is a left child of its parent Delete a non-leaf node A node that has 1 child A node that has 2 children Replaced by the largest element in its left subtree, or Replaced by the smallest element in its right subtree C.E., NCU, Taiwan Angela Chih-Wei Tang,

21 Deleting A Leaf Node From A Binary Search Tree C.E., NCU, Taiwan Angela Chih-Wei Tang,

22 Deleting A Non-leaf Node From A Binary Search Tree C.E., NCU, Taiwan Angela Chih-Wei Tang,