14. BinarySearchTrees_outside

# Binary search trees 6 search to search for a key k we

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Binary Search Trees 6 Search To search for a key k , we trace a downward path starting at the root The next node visited depends on the comparison of k with the key of the current node If we reach a leaf, the key is not found Example: get (4): Call TreeSearch(4,root) The algorithms for floorEntry and ceilingEntry are similar Algorithm TreeSearch ( k , v ) if v.isExternal () return v if k < v.key () return TreeSearch ( k , v.left ()) else if k = v.key () return v else { k > v.key () } return TreeSearch ( k , v.right ()) 6 9 2 4 1 8 < > = © 2010 Goodrich, Tamassia

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Binary Search Trees 7 Insertion To perform operation put (k, o), we search for key k (using TreeSearch) Assume k is not already in the tree, and let w be the leaf reached by the search We insert k at node w and expand w into an internal node Example: insert 5 6 9 2 4 1 8 6 9 2 4 1 8 5 < > > w w © 2010 Goodrich, Tamassia
Binary Search Trees 8 Deletion To perform operation erase ( k ), we search for key k Assume key k is in the tree, and let let v be the node storing k If node v has a leaf child w , we remove v and w from the tree with operation removeExternal ( w ), which removes w and its parent Example: remove 4 6 9 2 4 1 8 5 v w 6 9 2 5 1 8 < > © 2010 Goodrich, Tamassia

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Binary Search Trees 9 Deletion (cont.) We consider the case where the key k to be removed is stored at a node v whose children are both internal we find the internal node w that follows v in an inorder traversal we copy key ( w ) into node v we remove node w and its left child z (which must be a leaf) by means of operation removeExternal ( z ) Example: remove 3 3 1 8 6 9 5 v w z 2 5 1 8 6 9 v 2 © 2010 Goodrich, Tamassia
Binary Search Trees 10 Performance Consider an ordered map with n items implemented by means of a binary search tree of height h the space used is O ( n ) methods get , floorEntry , ceilingEntry , put and erase take O ( h ) time The height h is O ( n ) in the worst case and O (log n ) in the best case © 2010 Goodrich, Tamassia

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