1. 1 Joe Meehean

2. A A B B D D I I C C E E X X

3. A A B B D D I I C C E E X X  Terminology each circle is a node pointers are edges topmost node is the root bottom nodes are leaves  no outgoing edges  Every non-empty tree has one root one or more leaves

4.  Node A is the parent of node B  Node B is the child of node A  The root has no parent  All other nodes have exactly 1 parent A A B B

5.  Not a tree  D has 2 parents A A B B D D C C

6.  Path is a sequence of connected nodes  Length of a path is the number of edges in the path A to D is 2 C to F is 1 X is 0 A A B B D D C C F F X X Tree 1 Tree 2 Tree 3

7.  Height of a tree: length of its longest path from root to leaf  Ex: Height of 2 A A B B D D C C E E

8.  Depth of a node: length of path from root  Example A: 0 B: 1 E: 2 D: 2 C: 1 A A B B D D C C E E

9.  Subtrees of a node are the nodes rooted at a nodes children  As subtrees rooted at B rooted at C A A B B D D C C E E

10.  Special Tree  No node has more than 2 children can have 0,1, or 2  Each child is either “right” or “left” A A B B D D C C E E

11. A A B B D D C C E E X X Y Y Z Z S S T T

12. template class BinaryTreenode { private: D data_; BinaryTreenode * left_; BinaryTreenode * right_; public: BinaryTreenode(D d = D(), BinaryTreenode * left = NULL, BinaryTreenode * right = NULL); };

13.  Iterate through all nodes each node visited once, to… print all values see if a node has some property modify the node, etc…  4 common orders for visiting nodes preorder postorder in order (binary trees only) level order

14.  Depth-first traversal  Visit the root first  Recursive definition visit the root do a preorder traversal of each subtree, left to right  Ex: A B E D C F A A B B D D C C E E F F 1 2 34 5 6

15.  Depth-first traversal  Visit the root last  Recursive definition do a postorder traversal of each subtree, left to right visit the root  Ex: E D B F C A A A B B D D C C E E F F 6 3 12 5 4

16.  Depth-first traversal  For binary trees only  Visit root in between subtrees  Recursive definition in-order traversal of left subtree visit the root in-order traversal of right subtree  Ex: E B D A F C A A B B D D C C E E F F 4 2 13 6 5

17.  Visit all nodes at level 1 (depth 1) then level 2, level 3, etc… always left to right (or some order)  Use a queue instead of recursion (implicitly uses a stack) q.push(root); while( !q.empty() ){ //dequeue node n //visit n //enqueue all of n’s children(L to R) }

18.  Special kind of binary tree  Each node stores a key sometimes an associated value  For each node n all keys in n’s left subtree are < key at n all keys in n’s right subtree are > key at n if duplicate keys allowed  keys that equal n can go left XOR right (not both) 18

19.  Insert a key (and associated data)  Lookup a key (and associated data)  Remove a key (…)  Print all keys in sorted using an inorder traversal 19

20. 6 6 4 4 5 5 9 9 2 2 6 6 4 4 7 7 9 9 2 2 Yes In order traversal produces: 2 4 5 6 9 No: 7 is not < 6 4 2 13 5 20

21. 21 6 6 4 4 5 5 9 9 3 3 lookup(2 ) 7 7 15 2 2

22. 22 6 6 4 4 5 5 9 9 3 3 7 7 15 lookup(2 ) 2 2 Eliminates half the nodes at every compare

23. 23 6 6 4 4 5 5 9 9 3 3 7 7 15 lookup(2 ) 2 2 Eliminates half the nodes at every compare

24. 24 6 6 4 4 5 5 9 9 3 3 7 7 15 lookup(2 ) 2 2 Eliminates half the nodes at every compare

25.  Worst-case O(height of tree)  Worst of worst height is N lookup is O(N)  Best worst-case height is log 2 N lookup is O(logN)  O(LogN) is waaaay better than O(N) 25

26.  New values inserted as leaves  Must choose position to respect BST ordering and to ensure we can find it with a lookup  Duplicate keys are not allowed 26

27.  Traverse the tree like a lookup  If we find a duplicate return an error  If we end up at a null (child of a leaf) make a new node with the key make it the child of the leaf  Note the above two were our base cases for lookup too 27

28.  Similar to lookup worst-case follow path from root to leaf O(logN) for a balanced tree O(N) for a completely unbalanced tree 28

29.  Find the node n w/ key to be deleted  Different actions depending on n’s # of kids Case 1: n has 0 kids (it’s a leaf)  set parent’s n-pointer (left or right) to null Case 2: n has 1 kid  set parent’s n-pointer to point to n’s only kid Case 3: n has 2 kids  replace n’s key with a key further down in the tree  delete that node 29

30. 8 8 20 16 15 18 17 … … … … delete(17) 30

31. 8 8 20 16 15 18 … … … … 31 delete(17) 17

32. 8 8 20 16 15 18 17 … … … … delete(16) 32

33. 16 8 8 20 15 18 17 … … … … delete(16) 33

34. 8 8 20 16 15 18 17 … … … … Smallest value in right subtree delete(15) 34

35. 8 8 20 16 18 17 … … … … 35 Case 2: 1 kid Replace 16 with it’s only child delete(15)

36. 8 8 20 16 18 17 … … … … 36 Case 2: 1 kid Replace 16 with it’s only child delete(15)

37.  Worst-case (delete root)  Root to leaf  H = Height of Tree O(H)  Balanced tree O(logN) where N is keys in tree  O(N) for a completely unbalanced tree 37

38.  DSW algorithm transform BST into a linked list transform linked list into balanced BST uses tree rotations  Tree rotation rotates a part of the tree makes a child the parent makes a parent the child preserves BST property 38

39.  Left-child tree rotation: (4 & 6) 39 6 6 4 4 5 5 9 9 2 2

40.  Left-child tree rotation: (4 & 6) 40 6 6 4 4 5 5 9 9 2 2 6 6 4 4 5 5 9 9 2 2 Make 4 parent of 6

41.  Left-child tree rotation: (4 & 6) 41 6 6 4 4 5 5 9 9 2 2 6 6 4 4 5 5 9 9 2 2 Assign 4’s abandoned child to 6.

42.  General left-child tree rotation 42 A A B B X Y Z k2 k1

43.  General left-child tree rotation 43 A A B B A A B B X Y Z Z X Y k2 k1 k2 k1

44.  General left-child tree rotation 44 A A B B A A B B X Y Z X Y Z k2 k1 k2 k1

45.  General left-child tree rotation 45 A A B B A A B B X Y Z X Y Z k2 k1 k2 k1

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