Hi Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
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1 Trees Hi Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
2 Tree node (13) degree of a tree (3) height ht of a tree (4) degree of a node leaf (terminal) Nonterminal parent children Sibling ancestor 3 Root Level A Subtree 1 B C D E F G H I J ancestor K L leaf M leaf 4 Trees 2
3 Terminology Degree of a node Number of subtrees of the node The node with degree 0 is a leaf or terminal node A node that has subtrees is the parent of the roots of the subtrees The roots of these subtrees are the children of the node. Children of the same parent are siblings. The ancestors of a node are all the nodes along the path from the root to the node. Trees 3
4 Representation of a tree List representation ( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) ) The root comes first, followed by a list of sub-trees data child 1 child 2... child n A B C D E F G H I J K L M Trees 4
5 Binary tree Consists of a root and two disjoint binary trees called the left subtree and the right subtree. Any tree can be transformed into binary tree by left child- right sibling representation A Left child-right sibling representation B C D E F G H I J K L M Trees 5
6 Left child-right sibling representation B A left child data right sibling E C K F G D L H M I J Trees 6
7 Samples of binary tree Skewed binary Tree Complete binary tree A A A B B B C C D E F G D E H I Trees 7
8 Full binary tree Afullbinarytreeofdepthk k is a binary tree of depth k having 2 k -1 nodes, k>=0. A B C D E F G H I J K L M N O Trees 8
9 Linked representation of a binary tree typedef struct treenode { ; char data; treenode *leftchild, *rightchild; data leftchild data rightchild leftchild rightchild Trees 9
10 Binary tree transversal Let L, V, and R stand for moving left, visiting the node, and moving right There are six possible combinations of traversal LVR, LRV, VLR, VRL, RVL, RLV Adopt convention that t we traverse left before right, only 3 traversals remain LVR, LRV, VLR inorder, postorder, preorder Trees 10
11 Arithmetic representation by binary tree A + * E * D / C B inorder traversal A / B * C * D + E infix expression preorder traversal + * * / A B C D E prefix expression postorder traversal AB/C*D*E * + postfix expression level order traversal + * E * D / C A B Null Trees 11
12 Inorder tree transversal (using recursive call) void inorder(treenode *ptr) { if (ptr) { inorder(ptr->leftchild); printf( %c, ptr->data); inorder(ptr->rightchild); A/B* C * D+E Trees 12
13 Preorder tree transversal (using recursive call) void preorder(treenode *ptr) { if (ptr) { printf( %c, ptr->data); preorder(ptr->leftchild); preorder(ptr->rightchild); + * * / A B C D E Trees 13
14 Postorder tree transversal (using recursive call) void postorder(treenode *ptr) { if (ptr) { postorder(ptr->leftchild); postorder(ptr->rightchild); printf( %c, ptr->data); AB/C*D*E+ * Trees 14
15 Iterative inorder traversal (using stack) treenode stack[max_stack_size]; int top; void iterinorder(treenode *node) { top= -1; /* initialize stack */ for (;;) { for (; node; node=node->leftchild) push(node); /* add to stack */ node= pop(); /* delete from stack */ if (!node) break; /* all nodes are traversed */ printf( %c, node->data); node = node->rightchild; Trees 15
16 Level order traversal (using queue) treenode queue[max_queue_size]; _ int front, rear; void levelorder(treenode l d *ptr) { front = rear = 0; if (!ptr) return; /* empty tree */ addq(ptr); for (;;) { ptr = deleteq(); + * E * D / C A B Trees 16
17 if (ptr) { printf( %c, ptr->data); if (ptr->leftchild) addq(ptr->leftchild); if (ptr->rightchild) addq(ptr->rightchild); else break; Trees 17
18 Copy binary trees treenode* copy(treenode *original) { treenode *temp; if (original) { temp=(treenode *) malloc(sizeof(treenode)); temp->leftchild=copy(original->leftchild); temp->rightchild=copy(original->rightchild); >rightchild); temp->data=original->data; return temp; return NULL; Trees 18
19 Test equality of binary trees int equal(treenode *first, treenode *second) { return ((!first &&!second) ( first && second && (first->data == second->data) && equal(first->leftchild, second->leftchild) && equal(first->rightchild >rightchild, second->rightchild))) Trees 19
20 Threaded binary tree Two many null pointers in current representation of binary trees n: number of nodes total links: 2n If number of non-null links is n-1, the null links: 2n-(n-1)=n+1 Replace these null pointers with some useful threads If ptr->leftchild is null, replace it with a pointer to the node that t would be visited before ptr in an inorder traversal If ptr->rightchild is null, replace it with a pointer to the node that would be visited after ptr in an inorder traversal Trees 20
21 A threaded binary tree root A dangling B C dangling D E F G H I inorder traversal: H, D, I, B, E, A, F, C, G Trees 21
22 Data structure for threaded binary tree typedef struct threadedtree { ; short leftthread; threadedtree *leftchild; char data; threadedtree *rightchild; short rightthread; Trees 22
23 A empty pythreaded binary tree Header node leftthread leftchild data rightchild rightthread TRUE FALSE Initial condition: TRUE FALSE: rightchild point to leftchild serves as a thread the head node leftthread is set to FALSE when a new node is added d Trees 23
24 Memory representation of threaded tree root Header node f -- f f A f f B f f C f f D f t E t t F t t G t t H t t I t f (FALSE): leftchild or rightchild point a node t (TRUE): leftchild or rightchild serves as a thread Trees 24
25 Finding the inorder successor of a node threadedtree* insucc(threadedtree *tree) { threadedtree *temp; temp = tree->rightchild; if (!tree->rightthread) /* if rightchild is not a thread */ while (!temp->leftthread) temp = temp->leftchild; return temp; insucc(c) F Insucc(E) A insucc(root) H Trees 25
26 Inorder traversal of a threaded tree void tinorder(threadedtree *tree) { threadedtree *temp = tree; for (;;) { temp = insucc(temp); if (temp==tree) break; printf( %3c, temp->data); tinorder(root) H D I B E A F C G Trees 26
27 Binary search tree Every element has a unique key The keys in a nonempty left subtree are smaller than the key in the root of subtree The keys in a nonempty right subtree are larger than the key in the root of subtree The left and right subtrees are also binary search trees Trees 27
28 Binary search tree vs. linear search array Linear search array 50 Binary search tree 15 items Worst case: search 15 times Only 4 levels if can be represented by a fully binary tree 50 Worst case: search 4 times Trees 28
29 Searching a key in a binary search tree (recursive) treenode* search(treenode *root, int key) { if (!root) return NULL; if (key == root->data) t return root; if (key < root->data) return search(root->leftchild,key); return search(root->rightchild,key); Trees 29
30 Searching a key in a binary search tree (iterative) treenode* itersearch(treenode *tree, int key) { while (tree) { if (key == tree->data) return tree; if (key < tree->data) tree = tree->leftchild; else tree = tree->rightchild; return NULL; Trees 30
31 Insert node into a binary search tree Insert 80 Insert 35 Trees 31
32 Insert node into a binary search tree treenode* insertnode(treenode *tree, int num){ treenode *ptr; treenode *temp = modifiedsearch(tree, num); if (temp!tree) { ptr = (treenode *) malloc(sizeof(treenode)); ptr->data = num; /* create a new node */ ptr->leftchild = ptr->rightchild = NULL; if (tree) /* add the new node to the child of temp*/ if (num<temp->data) temp->leftchild = ptr; else temp->rightchild = ptr; else /* tree is empty, so a new tree with one node is created*/ tree = ptr; return tree; Trees 32
33 Modified search treenode* modifiedsearch(treenode *tree, int key) { treenode* lastnode; If tree is empty or key is if (!tree) return NULL; present, it return NULL while (tree) { if (key == tree->data) return NULL; lastnode = tree; if (key < tree->data) tree = tree->leftchild; else tree = tree->rightchild; return lastnode; Otherwise, return the last node that was encountered Trees 33
34 Deletion form a binary search tree Deleting a leaf node is quite easy Deleting a non-leaf node 40 Deleting Search smallest in the right Search largest in the left Trees 34
35 Reference Ellis Horowitz, Sartaj Sahni, and Susan Anderson-Freed. Fundamentals of Data Structures in C, Silicon Press Trees 35
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