1. Lecture 91 Data Structures, Algorithms & Complexity Insertion and Deletion in BST GRIFFITH COLLEGE DUBLIN

2. Lecture 92 Introduction Insertion and Deletion from a binary search tree cause the dynamic set to change The data structure must be modified to reflect this while maintaining the search tree property Insertion proves to be a relatively straightforward operation Deletion as we will see is slightly more complicated

3. Lecture 93 Insertion Take the following tree ask how would we insert a node with key 13 into the tree? 12 17 5 18 1915 92

4. Lecture 94 Insertion The easiest way is to start at the root and travel down the tree until you find a leaf position at which the insertion takes place 12 17 5 18 1915 92 13

5. Lecture 95 Insertion Algorithm Tree-Insert(T, z) y = nil x = root[T] while x <> nil do y = x if key[z] < key[x] thenx = left[x] elsex = right[x] endwhile parent[z] = y if y = nil then root[T] = z else if key[z] < key[y] thenleft[y] = z elseright[y] = z endif endalg

6. Lecture 96 Discussion The variable x is used to trace the path down the tree The variable y always holds a reference to the parent of x This is needed to splice x in at the correct position in the tree Tree insert will at worst have to travel to the furthest leaf on the tree, i.e. the height of the tree Therefore, tree insert runs in O(h) time

7. Lecture 97 Deletion The procedure for deleting a given node z from a binary search tree will take as an argument a reference to a node (z) The procedure needs to consider three cases If z has no children simply remove it If the node has only a single child, we “splice out” z by making a new link between its child and its parent If the node has two children, we splice out z’s successor y, which has at most one child, and then replace the contents of z with the contents of y

8. Lecture 98 Deletion of a leaf If z has no children, simply delete it 1515 5 123 10 6 7 16 20 18231313 z 1515 5 123 10 6 7 16 20 1823

9. Lecture 99 Deletion where z has one child 1515 5 123 10 6 7 16 20 18231313 z 1515 5 123 10 6 7 20 1823 1313

10. Lecture 910 Deletion where z has two children 1515 5 123 10 6 7 16 20 18231313 z y 1515 6 123 10 7 16 20 18231313 z

11. Lecture 911 Algorithm Tree-Delete(T, z) 1if left[z] = nil or right[z] = nil then y = z 2else y = Tree-Successor(z) 3if left[y] <> nil then x = left[y] 4else x = right[y] 5if x <> nil then parent[x] = parent[y] 6if parent[y] = nil then 7root[T] = x 8else if y = left[parent[y]] then 9left[parent[y]] = x 10else right[p[y]] = x 11if y <> z then key[z] = key[y] 12(if y has other fields, copy them also) 13return y endalg

12. Lecture 912 Discussion In lines 1 – 2 the algorithm dermines a node y to splice out. This is either the input node z, if z has at most 1 child, or the successor of z, if z has two children In lines 3-4, x is set ot the non-nil child of y, or to nil if y has no children In lines 5-10 the node y is spliced out by modifying the references parent[y] and x Here the boundary conditions when x = nil or when y is the root need to be dealt with

13. Lecture 913 Discussion In lines 11-12 if the successor of z was the node spliced out, the contents of z are moved from y to z, overwriting the previous contents The node y is returned in line 13 for use in the calling routine The procedure runs in O(h) time on a tree of height h The deletion algorithm requires the Tree-Successor algorithm

14. Lecture 914 Tree-Successor In a BST, if all keys are distinct, the successor of a node x is the node with the smallest key greater than key[x] The structure of a binary search tree allows us to determine the successor of a node without ever comparing keys The following algorithm returns the successor of a node x in a BST if it exists and nil if x has the largest key in the tree

15. Lecture 915 Tree-Successor Tree-Successor(x) if right[x] <> nil then return Tree-Minimum(right[x]) endif y = parent[x] while y <> nil and x = right[y] do x = y y = parent[y] endwhile return y endalg

16. Lecture 916 Discussion The code is broken into two cases If the right subtree of node x is nonempty, then the successor of x is just the left-most node in the right subtree, which found by calling Tree-Minimum(right) If the right subtree of x is empty and x has a successor y, then y is the lowest ancestor of x whose left child is also an ancestor of x, or if x is the left child of its parent, then the parent is the successor To find y we simply go up the tree from x until we encounter a node that is the left child of its parent

17. Lecture 917 Tree Successor Here successor of 13 is 15, 12 is the successor of 10 and 10 is the successor of 7 1515 5 123 10 6 7 16 20 18231313

18. Lecture 918 Balancing Binary Search Trees can be seen to be very efficient data structures However, ifhte tree is changed by a series of insertions and deletions then we can end up with an unbalanced tree The efficiency of such a tree may be no better than that of a linked list There are a number of ways we can modify BSTs to maintain efficiency by maintaining balance In the next lecture we will look at one of these

19. Lecture 919 Summary Insertion into and deletion from Binary Search trees must maintain the Binary Search Tree property i.e. 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 node in the right subtree of x, then key[x]  key[y] Insertion involves travelling down the tree until we find the position to splice the new node into Deletion is more complex and involves finding a successor node Insertion and deletion can cause the BST to become unbalanced so leading to a decrease in efficiency