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Finally, trees can also be traversed in level-order, where we visit every node on a level before going to a lower level. This is also called Breadth-first traversal

Once the binary search tree has been created, its elements can be retrieved in order by recursively traversing the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed. The tree may also be traversed in pre-order or post-order traversals. The following is the implementation of these traversals:

preorder(node)

print node.value

if node.left ≠ null then preorder(node.left)

if node.right ≠ null then preorder(node.right)

inorder(node)

if node.left ≠ null then inorder(node.left)

print node.value

if node.right ≠ null then inorder(node.right)

postorder(node)

if node.left ≠ null then postorder(node.left)

if node.right ≠ null then postorder(node.right)

print node.value

All three sample implementations will require stack space proportional to the height of the tree. In a poorly balanced tree, this can be quite considerable.

5.2.5. sort

(From Wikipedia, the free encyclopedia)

A binary search tree can be used to implement a simple but inefficient sorting algorithm . Similarly to heapsort , we insert all the values we wish to sort into a new ordered data structure — in this case a binary search tree — and then traverse it in order, building our result:

def build_binary_tree(values):

tree = None

for v in values:

tree = binary_tree_insert(tree, v)

return tree

def traverse_binary_tree(treenode):

if treenode is None: return []

else:

left, value, right = treenode

return (traverse_binary_tree(left) + [value] + traverse_binary_tree(right))

The worst-case time of build_binary_tree is Θ(n2) — if you feed it a sorted list of values, it chains them into a linked list with no left subtrees. For example, build_binary_tree([1, 2, 3, 4, 5]) yields the tree (None, 1, (None, 2, (None, 3, (None, 4, (None, 5, None))))).

There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree . If this same procedure is done using such a tree, the overall worst-case time is O (nlog n), which is asymptotically optimal for a comparison sort . In practice, the poor cache performance and added overhead in time and space for a tree-based sort (particularly for node allocation ) make it inferior to other asymptotically optimal sorts such as quicksort and heapsort for static list sorting. On the other hand, it is one of the most efficient methods of incremental sorting, adding items to a list over time while keeping the list sorted at all times.

5.3. types of binary search trees

(From Wikipedia, the free encyclopedia)

There are many types of binary search trees. AVL trees and red-black trees are both forms of self-balancing binary search trees . A splay tree is a binary search tree that automatically moves frequently accessed elements nearer to the root. In a treap ("tree heap "), each node also holds a priority and the parent node has higher priority than its children.

5.3.1. performance comparisons

D. A. Heger (2004) presented a performance comparison of binary search trees. Treap was found to have the best average performance, while red-black tree was found to have the smallest amount of performance fluctuations.

5.3.2. optimal binary search trees

If we don't plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct an optimal binary search tree, which is a search tree where the average cost of looking up an item (the expected search cost) is minimized.

Assume that we know the elements and that for each element, we know the proportion of future lookups which will be looking for that element. We can then use a dynamic programming solution, detailed in section 15.5 of Introduction to Algorithms by Thomas H. Cormen Sec Edition, to construct the tree with the least possible expected search cost.

Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if you have a BST of English words used in a spell checker , you might balance the tree based on word frequency in text corpuses, placing words like "the" near the root and words like "agerasia" near the leaves. Such a tree might be compared with Huffman trees , which similarly seek to place frequently-used items near the root in order to produce a dense information encoding; however, Huffman trees only store data elements in leaves and these elements need not be ordered.

If we do not know the sequence in which the elements in the tree will be accessed in advance, we can use splay trees which are asymptotically as good as any static search tree we can construct for any particular sequence of lookup operations.

Alphabetic trees are Huffman trees with the additional constraint on order, or, equivalently, search trees with the modification that all elements are stored in the leaves. Faster algorithms exist for optimal alphabetic binary trees (OABTs).

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
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Source:  OpenStax, Data structures and algorithms. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10765/1.1
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