Chapter 9: Search Trees I

• find(k) - Return the position of an item with key k, and return a null position if no such item exists.
• findAll(k) - Return an iterator of positions for all items with key k, and return a null position if no such item exists.
• insertItem(k,e) - Inserts an item with element e and key k.
• removeElement(k) - Remove an item with key k. An error condition occurs if there is no such item.
• removeAllElements(k) - Remove all items with key equal to k.

9.1 Binary Search Trees

• A binary search tree is a binary tree storing keys (or key-element pairs) at its internal nodes and satisfying the following property:
• Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v.
  We have key(u) ≤ key(v) ≤ key(w)
• External nodes do not store items
• Inorder traversal visits the keys of its dictionary in nondecreasing order.

• To search for a key k, we trace a downward path starting at the root
• The next node visited depends on the outcome of the comparison of k with the key of the current node
• If we reach a leaf, the key is not found and we return a null position

• Example: find(4)

Analysis of Binary Tree Searching

• Function find(k) on dictionary D runs O(h) time, where h is the height of the binary search tree T used to implement D.

9.1.2 Update Operations

• To perform operation insertItem(k, e), we search for key k
• 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

• To perform operation removeElement(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 removeAboveExternal(w)
• Example: remove 4

• 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 removeAboveExternal(z)
• Example: remove 3

• Consider a dictionary with n items implemented by means of a binary search tree of height h
• the space used is O(n)
• methods findElement(), insertItem() and removeElement() take O(h) time
• The height his O(n) in the worst case and O(log n) in the best case

9.1.3 C++ Implementation