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12.Binary Search Trees

Binary Search Trees (BSTs) are data structures that maintain sorted order, enabling efficient querying and manipulation of data. This overview covers the essential properties of BSTs, including the binary search property, traversal methods such as inorder, preorder, and postorder, and dynamic-set operations like searching, insertion, and deletion. Key algorithms for finding minimum and maximum values, as well as successors and predecessors, are also discussed, highlighting their time complexities. Master these concepts to leverage BSTs effectively in computational applications.

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12.Binary Search Trees

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  1. 12.Binary Search Trees Hsu, Lih-Hsing

  2. 12.1 What is a binary search tree? • Binary-search property: Let x be a node in a binary search tree. If y is a node in the left subtree of x, then key[y]  key[x]. If y is a node in the right subtree of x, then key[x]  key[y].

  3. Binary search Tree

  4. Inorder tree walk INORDER_TREE_WALK(x) 1 if 2 then INORDER_TREE_WALK(left[x]) 3 print key[x] 4 INORDER_TREE_WALK(right[x])

  5. Theorem 12.1 If x is the root of an n-node subtree, then the call INORDER-TREE-WALK(x) takes (n) time.

  6. Preorder tree walk • Postorder tree walk

  7. 12.2 Querying a binary search tree

  8. TREE_SEARCH(x,k) TREE_SEARCH(x,k) 1 ifor 2 then returnx 3 if 4 then return TREE_SEARCH(left[x],k) 5 else return TREE_SEARCH(right[x],k)

  9. ITERATIVE_SEARCH(x,k) ITERATIVE_SEARCH(x,k) 1 While or 2 do if 3 then 4 then 5 returnx

  10. MAXIMUM and MINIMUM • TREE_MINIMUM(x) 1 while left[x]  NIL 2 dox  left[x] 3 returnx • TREE_MAXIMUM(x) 1 while right[x]  NIL 2 dox  right[x] 3 returnx

  11. SUCCESSOR and PREDECESSOR

  12. TREE_SUCCESSOR TREE_SUCCESSOR 1 if 2 then returnTREE_MINIMUM(right[x]) 3 4 whileand 5 do 6 7 returny

  13. Theorem 12.2 • The dynamic-set operations, SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR can be made to run in O(h) time on a binary search tree of height h.

  14. 12.3 Insertion and deletion

  15. Insertion Tree-Insert(T,z) 1 y  NIL 2 x root[T] 3 whilex NIL 4 doy x 5 ifkey[z] < key[x] 6 thenx left[x] 7 elsex right[x] 8 p[z]  y

  16. 9 ify = NIL 10 thenroot[T]  z tree T was empty 11 elseifkey[z] < key[y] 12 thenleft[y]  z 13 elseright[y]  z

  17. Inserting an item with key 13 into a binary search tree

  18. Deletion Tree-Delete(T,z) 1 ifleft[z] = NILorright[z] = NIL 2 theny z 3 elsey Tree-Successor(z) 4 if left[y]  NIL 5 thenx left[y] 6 elsex right[y] 7 ifx NIL 8 thenp[x]  p[y]

  19. 9 ifp[y] = NIL 10 thenroot[T]  x 11 else ify = left[p[y]] 12 thenleft[p[y]]  x 13 elseright[p[y]]  x 14 ify z 15 thenkey[z]  key[y] 16 copy y’s satellite data into z 17 returny

  20. z has no children

  21. z has only one child

  22. z has two children

  23. Theorem 12.3 • The dynamic-set operations, INSERT and DELETE can be made to run in O(h) time on a binary search tree of height h.

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