Binary Search Trees BINARY SEARCH TREES Important data

Binary Search Trees

BINARY SEARCH TREES Important data structure for dynamic sets – dictionary or priority queue. Many BST operations are performed in O(h) where h is height of tree. Node in tree is a record that can have 5 fields : pointer to parent node, pointer to left/right child, key and data.

Quick Review Root? Leaf? Interior Node? Parent/child? Sub-tree? This nearly complete BST is well balanced –> log n runtimes.

Another well balanced example.

An unbalanced BST giving linear runtimes.

BINARY SEARCH TREE PROPERTY: 1. if y is left sub-tree of x then key[y] <= key[x]. 2. if y is right sub-tree of x then key[y] > key[x]. Pulling keys out of tree in sorted order requires an INORDER walk of the tree.

Inorder. Tree. Walk(x) //start at root if x != null then Inorder. Tree. Walk(Left[x]) print(key[x]) Inorder. Tree. Walk(Right[x]) Show sequence of output using example tree. ABDFHK. Do walk. Runtime of tree walk with an n-node BST ?

Other operations : Searching and Insertion. Where is minimum? maximum? Tree. Search(x, k) //x starts at root, k is key if (x = null) or (k = key[x]) then return x else if k < key[x] then return Tree. Search(left[x], k) else return Tree. Search(right[x], k) Trace algorithm by searching for 4 and 10 in example tree. If k is not in tree then return nil.

Insertion of x in a BST: code is similar to search code above. It is modified by placing x at a leaf in the proper sub-tree. The code uses a trailing back-pointer to keep track of parent of current node (same idea as inserting into a linked list). Trace algorithm by inserting 18 in tree. Runtime of Insertion = _______?

Do BST Activity in pairs

BST OPERATIONS Minimum (maximum) is a required operation for priority queues. BST as a priority queue implements minimum as: Tree. Minimum(x) //x is pointer to root while left[x] != null then x = left[x] return x Runtime for tree minimum ?

Successor operation retrieves successor node of x. Successor node of x is node with key that immediately follows the key of x, in sorted order. Tree. Successor(x) if right[x] != null then {if right, suc. is min in subtree} return Tree. Minimum(right[x]) y = parent[x] while(y!=null) and(x=right[y]) do {else suc. is back up x=y tree to left of parent} y =parent[y] return y Do an example trace of algorithm Runtime of Tree. Successor = ____? Note that predecessor algorithm is analogous.

Deletion operation removes a node from the tree. The following is a very informal pseudo code for operation. Tree. Delete(T, x) if x has no children then {case 0} remove x if x has one child then {case 1} make parent[x] point to child remove x if x has two children then {case 2] swap x with its successor perform case 0 or 1 to delete it

Do example trace for three cases of algorithm. See pseudo code next slide. Note: the above swap could also be done with predecessor. Runtime of Delete operation ?

D or A

Minimizing runtimes: • Problem: the worst case for operations on BST are ______, which is no better than using a linked list. • Solution: guarantee that the BST is balanced which minimizes its height. Method: restructure tree when necessary. Nothing special required for query operations but extra work needed when tree is modified via an insert or delete operation.

Refer to Pointer Example in BSTree Code – run code and discuss. This shows that passing a pointer by value and passing a pointer by reference have a subtle difference. Simple query operations on a BST can be accomplished by passing a pointer by value. Operations that modify the tree – insert, remove, rebalance, must use the pass a pointer by refernce mode.