1. Chapter 10 Search Trees 10.1 Binary Search Trees Search Trees
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2. 10.1 Binary Search Trees < > = 6 2 9 1 4 8 Dr Zeinab Eid

3. 10.1 Binary Search Trees
A binary search tree is a proper binary tree storing keys (or key-value entries) 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
An inorder traversal of a binary search trees visits the keys in increasing order 6 9 2 4 1 8
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Search Trees

4. Search
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 null
Example: find(4):
Call TreeSearch(4,root)
Algorithm TreeSearch(k, v)
if T.isExternal (v) then
return v
if k < key(v)
return TreeSearch(k, T.left(v))
else if k = key(v)
else { k > key(v) }
return TreeSearch(k, T.right(v))
< 6 2
> 9 1 4 = 8
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Search Trees

5. 10.1.2 Update Operations Insertion Removal Dr Zeinab Eid Dr Zeinab Eid
Search Trees

6. Insertion
Let us assume a proper binary tree T supports the following update operation:
insertAtExternal(v,e):Insert the element e at the external node v, and expand v to be internal, having new (empty) external node children; an error occurs if v is an internal node.
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Search Trees

7. Insertion
< 6
To perform operation insert(k, e), 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 2 9
> 1 4 8
> w 6 2 9 1 4 8 w 5
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Search Trees

8. Removal
We assume that a proper binary tree T supports the following additional update operation:
removeExternal(v): Remove an external node v and its parent, replacing v’s parent with v’s sibling; an error occurs if v is not external.
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Search Trees

9. Removal < To perform operation remove(k), we search for key k: >6
<
To perform operation remove(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 2 9
> v 1 4 8 w 5 6 2 9 1 5 8
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Search Trees

10. Removal (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 1 v 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9
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Search Trees

11. Example: Delete 65: - Find In-order Successor of 65, which is 76.
- Copy the key of this successor node into node to be deleted.
- Remove this successor node, which must have external node as its left child.
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Search Trees

12. Performance Analysis of Binary-Search Tree
All Search, insert and remove algorithms traverse a path from the root to a node in the BST.
In all three algorithms, we spend O(1) time at each node visited.
In the worst case, the number of nodes visited is proportional to the height h of T.
Thus, in a BST of n-nodes, all search, insert and remove methods run in O(h) time.
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Search Trees

13. Performance Analysis of Binary-Search Tree
A BST, T, is an efficient implementation of a dictionary of n-entries only if the height of T is small.
Best Case: T has height h=log(n+1), leading to a logarithmic-time performance of all dictionary operations.
Worst Case: T has height n, leading to a linear-time performance, as in case of a set of ordered keys.
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Search Trees

14. Performance
Consider a dictionary with n items implemented by means of a binary search tree of height h
the space used is O(n)
methods search, insert and remove take O(h) time
The height h is O(n) in the worst case and O(log n) in the best case
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Search Trees

15. Notice That:
The running time of search and update operations in a BST varies dramatically, depending on the tree’s height.
On the average, a BST with n-keys, generated by a random series of insertions and removals, of n keys has an expected height of O(log n) .
Nevertheless, we must not use standard BST in applications, where updates are not random.
Dr Zeinab Eid
Dr Zeinab Eid
Search Trees