Binary Search Trees. July 13th, Computer Information Systems. c 2009 Vaidė Narváez

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1 Binary Search Trees Vaidė Narváez Computer Information Systems July 13th, 2010

2 Introduction Dynamic set operations: Search, Minimum, Maximum Predecessor, Successor Insert, Delete Time proportional to the height of the tree (h)

3 Binary Search Trees (BSTs) Data structure organized as a tree Represented, e.g., by a linked data structure Node fields: Key, satellite data, left, right, p

4 Binary Search Trees (BSTs) Data structure organized as a tree Represented, e.g., by a linked data structure Node fields: Key, satellite data, left, right, p

5 Binary Search Trees (BSTs) Data structure organized as a tree Represented, e.g., by a linked data structure Node fields: Key, satellite data, left, right, p Height of BST is the number of links from the root to the deepest node.

6 Binary search Trees (2) Binary-search tree property Let x be a node in a binary search tree. If y is a node in the left subtree of x, then y.key x.key. If y is a node in the right subtree of x, then x.key y.key.

7 Binary search Trees (2) Binary-search tree property Let x be a node in a binary search tree. If y is a node in the left subtree of x, then y.key x.key. If y is a node in the right subtree of x, then x.key y.key.

8 Exercise... For the set of keys {1,4,5,10,16,17,21}, draw binary search trees of height 2, 3, 4, 5 and 6.

9 Walking in a BST inorder tree walk (prints out keys in sorted order) preorder tree walk postorder tree walk INORDER-TREE-WALK(x) 1 if x = NIL 2 then INORDER-TREE-WALK(x.left) 3 print x.key 4 INORDER-TREE-WALK(x.right) It takes Θ(n) to walk an n-node BST

10 Exercises... Write a pseudo-code for a non recursive algorithm that performs an inorder tree walk. (Hint: there is an easy solution that uses a stack as an auxiliary data structure )

11 Exercises... Write a pseudo-code for a non recursive algorithm that performs an inorder tree walk. (Hint: there is an easy solution that uses a stack as an auxiliary data structure ) Write a pseudo-code for a recursive algorithm that performs preorder and postoder tree walk.

12 Searching in a BST TREE-SEARCH(x, k) 1 if x = NIL or k = x.key 2 then return x 3 if k < x.key 4 then return TREE-SEARCH(x.left, k) 5 else return TREE-SEARCH(x.right, k) run in O(h) time ITERATIVE-TREE-SEARCH(x, k) 1 while x = NIL and k = x.key 2 do if k < x.key 3 then x x.left 4 else x x.right 5 return x

13 Min/Max in a BST TREE-MINIMUM(x) 1 while x.left = NIL 2 do x x.left 3 return x TREE-MAXIMUM(x) 1 while x.right = NIL 2 do x x.right 3 return x run in O(h) time

14 Successor of a BST Successor of x is the node with the smallest key greater than x.key TREE-SUCCESSOR(x) 1 if x.right = NIL 2 then return TREE-MINIMUM(x.right) 3 y x.p 4 while y = NIL and x = y.right 5 do x y 6 y y.p 7 return y

15 Successor of a BST Successor of x is the node with the smallest key greater than x.key TREE-SUCCESSOR(x) 1 if x.right = NIL 2 then return TREE-MINIMUM(x.right) 3 y x.p 4 while y = NIL and x = y.right 5 do x y 6 y y.p 7 return y runs in O(h) time

16 Exercise...

17 Insertion TREE-INSERT(T, z) 1 y NIL 2 x T.root 3 while x = NIL 4 do y x 5 if z.key < x.key 6 then x x.left 7 else x x.right 8 z.p y 9 if y = NIL 10 then T.root z 11 else if z.key < y.key 12 then y.left z 13 else y.right z runs in O(h) time

18 Exercise...

19 Deletion (1) Node has no children: delete node with key 13

20 Deletion (2) Node has only a single child: delete node with key 16

21 Deletion (3) Node has two children: delete node with key 5

22 Deletion (4) TREE-DELETE(T, z) 1 if left[z] = NIL or right[z] = NIL 2 then y z 3 else y TREE-SUCCESSOR(z) 4 if left[y] = NIL 5 then x left[y] 6 else x right[y] 7 if x = NIL 8 then p[x] p[y] 9 if p[y] = NIL 10 then root[t ] x 11 else if y = left[p[y]] 12 then left[p[y]] x 13 else right[p[y]] x 14 if y = z 15 then key[z] key[y] 16 copy y satellite data into z 17 return y lines 1-3: determine node to splice out lines 4-6: x is set to child of y, or to NIL if y has no children lines 7-13: y is spliced out lines 14-16: move data

Algorithms, Spring 2014, CSE, OSU Lecture 4: Binary search trees

6331 - Algorithms, Spring 2014, CSE, OSU Lecture 4: Binary search trees Instructor: Anastasios Sidiropoulos January 15, 2014 Binary search trees For every node x: x.k : key x.p : pointer to the parent