### Binary Search Trees

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Binary Search Trees
Chapter 12
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Objectives
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The idea of Binary Search Tree
Operations of Binary Search Tree
The run time of Binary Search Tree
Analysis of Binary Search Tree
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Search trees -1
 Search trees are data structures that support many
dynamic-set
operations,
including
SEARCH,
MINIMUM, MAXIMUM, PREDECESSOR, SUCCESSOR,
INSERT, and DELETE.
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Search trees - 2
 Basic operations on a binary search tree take time
proportional to the height of the tree.
 For a complete binary tree with n nodes, such operations
run in” O(lg n) = height” worst-case time.
 If the tree is a linear chain “ linear time algorithm” of n
nodes, however, the same operations take O (n) worstcase time (the process will always cover all the nodes = O
(n)).
 The height of the Binary Search Tree equals the number of
links from the root node to the deepest node.
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What is a binary tree?
 Property 1: each node can have up to two successor nodes
(children)
 The predecessor node of a node is called its parent
 The "beginning" node is called the root (no parent)
 A node without children is called a leaf
 Property 2: a unique path exists from the root to every other
node
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What is a binary search tree?
 a tree can be represented by a linked data structure in
which each node is an object.
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key field - 1
 In addition to a key field and satellite data, each node
contains fields left, right, and p that point to the
nodes corresponding to its left child, its right child,
and its parent, respectively.
 If a child or the parent is missing, the appropriate
field contains the value NIL. The root node is the only
node in the tree whose parent field is NIL.
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key field -2
 The keys in a binary search tree are always stored in such a
way as to satisfy the binary-search-tree property:
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Inorder, preorder, and postorder
 The binary-search-tree property allows us to print out all
the keys in a binary search tree in sorted order by a simple
recursive algorithm, called an inorder tree walk. This
algorithm is so named because the key of the root of a
subtree is printed between the values in its left subtree and
those in its right subtree.
 Similarly, a preorder tree walk prints the root before the
values in either subtree, and
 A postorder tree walk prints the root after the values in its
subtrees.
 It takes O(n) time to walk a tree of n nodes
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INORDER-TREE-WALK procedure
 Procedure to print all the elements in a binary search
tree T , we call INORDER-TREE-WALK(root[T ]).
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Querying a binary search tree
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Search the BST x for a value k
To search for the key 13 in the tree, we follow
the path 15 → 6 → 7 → 13 from the root.
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Note: Search operation takes time O(h)
“proportional to the height of the tree”,
where h is the height of a BST
Minimum and maximum
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Note: max and min operations take time O(h)
“proportional to the height of the tree”,
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where
h is the height of a BST
Successor and predecessor
 Given a node in a binary search tree, it is sometimes
important to be able to find its successor in the
sorted order determined by an in-order tree walk. If
all keys are distinct, the successor of a node x is the
node with the smallest key greater than key[x].
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- If the right subtree of node x is nonempty, then the successor of x is just the
leftmost node in the right subtree,
- If the right subtree of node 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.
- the successor of the node with key 15 in Figure 12.2 is the node with key 17
- the successor of the node with key 13 is the node with key 15.
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Insertion and deletion
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Insertion and deletion operations take
time proportional to the height of the
tree O(h)
Deletion (in lab.)
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Randomly built binary search trees
Self study
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Conclusion -1
 Binary search trees come in many shapes. The shape of the
tree determines the efficiency of its operations.
 Logarithmic time is generally much faster than linear time.
For example, for n = 1,000,000: log2 n = 20.
 Using binary search trees to represent sets: insert, search, remove,
…………… O(h) = O(log2 n)
 Using linear data structures to represent sets: insert, search,
remove, …………… O(n)
 Balanced BTS:
 Easier insertion/deletion
 Running time = O(log2 n)
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Conclusion -2
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