1. Tree C and Data Structures Baojian Hua bjhua@ustc.edu.cn

2. What ’ s a Tree? book chap1chap2 chap3 sec11sec12sec21 subsec111

3. Definition of Tree A tree is a collection of nodes, which satisfies: there exists a unique root node r other nodes are classified into n (n>=0) disjoint sets T_1, …, T_n, and every T_i is also a tree. T_1, …, T_n are called sub-trees of r. Moral: Recursive definition Sub-trees disjoint

4. What ’ s a Tree? book chap1chap2 chap3 sec11sec12sec21 subsec111 unique root

5. Terminologies book chap1chap2 chap3 sec11sec12sec21 subsec111 node degree leaves root internal node parent & child siblings depth=1 depth=0 depth=2 depth=3

6. Binary Tree A binary tree is a collection of nodes, which satisfies: there exists a unique root node r other nodes are classified into n (0<=n<=2) disjoint ordered sets T_1, …, T_n, and every set T_i is also a binary tree. To note: 0<=Degree(any node)<=2 the order of sub-tree is relevant

7. Examples book chap1chap2 sec11 sec12sec21 subsec111 sec22 book chap1 sec11 subsec111

8. Properties of Binary Tree

9. Properties of Binary Tree

10. Full and Complete Binary Tree A full binary tree is a binary tree of depth k and 2^{k+1}-1 nodes each node either has degree 2 (internal nodes) or degree 0 (leaves) A n-node complete binary tree is a tree with nodes number of full binary tree

11. Abstract Data Types in C: Interface // in file “btree.h” #ifndef BTREE_H #define BTREE_H typedef struct btreeStruct *btree; btree newTree (); btree newTree2 (btree left, btree right, poly data); void insert (btree t, poly parent, poly child, char pos); void preOrder (btree t, void (*visit)(poly)); void inOrder (btree t, void (*visit)(poly)); void postOrder (btree t, void (*visit)(poly)); void levelOrder (btree t, void (*visit)(poly)); #endif

12. Implementation // in file “btree.c” #include “btree.h” struct btreeStruct { poly data; btree left; btree right; }; t leftdataright

13. Operations: “ new ” // “new” creates an empty new binary tree. Just // as the case for linked list, we may add a head // node. btree newTree () { btree t = (btree)malloc (sizeof (*t)); t->data = NULL; t->left = NULL; t->right = NULL; return t; } t leftdataright

14. Operations: “ new2 ” // “newTree2” creates an tree node with fields // properly initialized. btree newTree2 (btree left, btree right, poly data) { btree t = (btree)malloc (sizeof (*t)); t->data = data; t->left = left; t->right = right; return t; } t leftdataright

15. How to build a tree? book chap1chap2 sec11 sec12sec21 subsec111 sec22

16. How to build a tree? A tree can be built in a step-by-step way first new an empty tree t to insert edge (parent, child, ‘ l ’ ) or (parent, child, ‘ r ’ ), we write a function insert insert (t, parent, child, pos) puts the child node at the pos field of parent node in t if parent does not exist or has already had a pos filed, an error occurs

17. Operations: “ insert ” // a nearly correct version: btree insert (btree t, poly parent, poly child, char pos) { btree temp = search (t, parent); if (pos == ‘l’) { btree p = new2 (temp->left, NULL, child); temp->left = p; } else {} // similar case for pos == ‘r’ return t; }

18. Operations: “ search ” btree search (btree t, poly data) { btree p; if (t==NULL) return NULL; else if (t->data == data) // what’s “==“ ? return t; else { p = search (t->left, data); return (p) ? (p) : (search (t->right, data)); }

19. Client Code Create this tree: book chap1chap2 sec11 sec12sec21 subsec111 sec22 // the creation process: t = new (); t = insert (t, NULL, book, ‘ l ’ ); t = insert (t, book, chap1, ‘ l ’ ); t =insert (t, book, chap2, ‘ r ’ ); t=insert (t, chap1, sec11, ‘ l ’ ); t=insert (t, chap1, sec12, ‘ r ’ ); …

20. What ’ s wrong with “ insert ” ? Many subtle conditions: what if the search fail? what if the arguments parent or child be NULL? what if the pos field unequal to ‘ l ’ or ‘ r ’ ? … All these should be checked in real production code a style called defensive programming worth of great recommendation

21. Tree Traversal book chap1chap2 Traversal: a systematic way to visit all nodes in a tree. For instance, the visit strategies for the right tree may be: book, chap1, chap2 or: chap1, book, chap2 and so on. Next, we ’ ll study four of them, namely: pre-order, in- order, post-order and level-order.

22. Pre-order Traversal Pre-order: first visit the root node of the tree and then left subtree and finally right subtree Ex: book, chap1, chap2 book chap1chap2

23. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } } // try visit = printf t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

24. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // try visit = printf t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

25. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

26. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

27. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

28. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” // print out “d” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

29. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” // print out “d” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

30. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” // print out “d” // print out “c” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

31. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” // print out “d” // print out “c” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

32. Pre-order Traversal void preOrder (btree t, void (*visit)(poly)) { if (t) { visit (t->data); preOrder (t->left, visit); preOrder (t->right, visit); } // print out “a” // print out “b” // print out “d” // print out “c” t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

33. Moral preOrder is a recursive algorithm: defined on recursively defined data structures system (machine) keeps a stack to control the recursive order Generally, recursion are more elegant, easy- to-write and easy-to-reason than corresponding iterative ones A powerful programming idiom to recommend Some languages even encourage this by removing “ while ” and “ for ” completely

34. Level-order Traversal void levelOrder (btree t, void (*visit)(poly)) { queue q = newQueue (); if (t) enQueue (q, t); while (!queueIsEmpty(q)) { btree temp = deQueue (q); visit (temp->data); if (temp->left) enQueue (q, temp->left); if (temp->right) enQueue (q, temp->right); } return; }

35. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d”

36. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t

37. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->l print out “a”

38. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->r t->l print out “a”

39. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->r print out “a” print out “b”

40. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->l->r t->r print out “a” print out “b”

41. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->l->r print out “a” print out “b” print out “c”

42. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” t->l->r print out “a” print out “b” print out “c”

43. Example t left “a”“a” right /\ “b”“b” right/\ “c”“c” “d”“d” print out “a” print out “b” print out “c” print out “d”

44. Summary Tree is a non-linear data structure every element has 0, 1 or 2 successors structure could be inductively defined Operations could also be inductive defined recursive functions