ץע A. B C D E F G H E, B ( -.) - F I J K ) ( A :. : ע.)..., H, G E (. י : י.)... C,A,, F B ( 2

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Tobias Dorsey
5 years ago
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1 נת ני ני, 1

2 עץ E A B C D F G H I J K. E B, ( -.)F- )A( : ע :..)...H,G,E (. י י:.)...C,A,F,B ( 2

3 עץ E A B C D F G H I J K v : -,w w.v- w-.v :v ע. v- B- 3

4 ע E A B C D F G H I J K ע - v,1 B ( v-.)? A 4

5 E A B C D F G H I J K - v.)2 B ( v ע =..-1 5

6 A B C D -.(degree(f) = 3( E F G H I J K 6

7 עץ ינ י ע י י -,. A A B C D B D E F G H F G H I J K I K 7

8 Full Binary Tree ע י י -. A A B D B D E F G H E F G H I K K 8

9 Perfect Binary Tree - ע י י 9

10 Complete Binary Tree T ". 10

11 Dynamic Set (Dictionary) Elements have a key and satellite data Dynamic sets support queries such as: Search(S, k) Minimum(S) Maximum(S) Successor(S, x) Predecessor(S, x) Insert(S, x) Delete(S, x) 11

12 Binary Search Trees T.root points to the root of tree T In addition to satellite data, elements have: key: an identifying field inducing a total ordering left: pointer to a left child (may be null) right: pointer to a right child (may be null) p: pointer to a parent node (null for root) p key left right 12

13 Binary Search Trees Binary Search tree property: y.key x.key < z.key, for any nodes x, y and z, such that y in left sub tree of x and z in right sub tree of x Example: 13

14 Tree Walk Inorder tree walk: walk left visit root walk right Preorder tree walk: visit root walk left walk right Postorder tree walk: walk left walk right visit root inorder (node x) if null then inorder (x.left) print(x.key) inorder (x.right) Complexity: O(n), where n is the number of nodes of the tree 14

15 Tree Search search (node x, key k) if (x = null OR k = x.key) do return x; if (k < x.key) then return search(x.left, k) else return search(x.right, k) Complexity: O(h), where h is the height of the tree 15

16 Iterative Tree Search search(node x, key k) while (x null AND k x.key) do if (k < x.key) then x x.left else x x.right return x 16

17 Minimum, Maximum, Successor and Predecessor Minimum: return leftmost node in tree or null Maximum: return rightmost node in tree or null Successor (node x): if (x has a right subtree) then successor is minimum node in right subtree else successor is first ancestor of x whose left child is also ancestor of x Predecessor: symmetric to successor Complexity: O(h), where h is the height of the tree successor (node x) if.right null) then return minimum (x.right) y x.p while null AND =.right do x y y y.p return y 17

18 Tree Walk walk (node x) x minimum(x) while (x null) do print(x.key) x successor(x) Complexity: O(n), where n is the number of nodes of the tree 18

19 Insertion Adds an element x to the tree so that the binary search tree property continues to hold The basic algorithm Like the search procedure above Insert x in place of null Use a trailing pointer to keep track of where ou came from (like inserting into singly linked list) Complexity: O(h), where h is the height of the tree 19

20 Insertion insert (tree T, node z) y null x T.root while null) do y x if (z.key < x.key) then x x.left else x x.right z.p y if (y = null) then // T was empty T.root z else if (z.key < y.key) then y.left z else y.right z 20

21 3 cases: (1) x has no children: Remove x Deletion 21

22 3 cases: (1) x has no children: Remove x (2) x has one child: Splice out x Deletion 22

23 Deletion 3 cases: (1) x has no children: Remove x (2) x has one child: Splice out x (3) x has two children: Swap x with successor Perform case 1 or 2 to delete it Complexity: O(h), where h is the height of the tree 23

24 Deletion delete (tree T, node z) // Determine which node y to splice out. if (z.left = null OR z.right = null) then y z else y successor(z) // x is set preferably to a non-null child of y. if (y.left null) then x y.left else x = y.right // y is removed from the tree by manipulating pointers of y.p and x. if null) then x.p y.p if (y.p = null) then T.root x else if (y = (y.p).left) then (y.p).left x else (y.p).right x // If it was z s successor that was spliced out, copy its data into z. if z then z.key y.key and copy s satellite data into z 24

25 Summary All the basic operations in O(h) time, where h is the height of the tree. Worst case height n - 1. Degenerate tree Best case height lg n. Complete tree of size n 2 i nodes at each level i except maybe at last level. Hence, 2 h n. Hence, h lg n 25

26 Randomly built binary search tree The average height is much closer to the best case. Little is known about the average height when both insertion and deletion are used. Randomly Built Binary Search Tree Keys inserting in random order into an initially empty tree. Each of the n! permutations of the input keys is equally likely. Theorem: The average height of a randomlybuilt binary search tree is Θ(lg n). 26

Binary search trees (BST) Binary search trees (BST)

Binary search trees (BST) Binary search trees (BST) Tree A tree is a structure that represents a parent-child relation on a set of object. An element of a tree is called a node or vertex. The root of a tree is the unique node that does not have a parent