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The binary tree structure can be used as an efficient way to organize data objects that are totally ordered. This is done by maintaining the tree in such a way that for any given subtree, the data elements in its left subtree are less than the root and the data elements in the right subtree are greater than the root. Such a binary tree is called a binary search tree.

Binary search tree

1. binary search tree property (bstp)

Consider the following binary tree of Integer objects.

-7 |_ -55| |_ [] | |_ -16| |_ -20 | | |_ []| | |_ [] | |_ -9| |_ [] | |_ []|_ 0 |_ -4| |_ [] | |_ []|_ 23 |_ []|_ []

Notice the following property:

  • all elements in the left subtree are less than the root element,
  • and the root element is less than all elements in the right subtree.

Moreover, this property holds recursively for all subtrees.  It is called the binary search tree (BST) property.  

In general, instead of Integer objects, suppose we have a set of objects that can be compared for equality with "equal to" and "totally ordered" with an order relation called "less or equal to" .  Define "less than" to mean "less or equal to" AND "not equal to".  Let T be a BiTree structure that stores such totally ordered objects.  

Definition of binary search tree property

The binary search tree property (BSTP) is defined on the binary tree structure as follows.

  • An empty binary tree satisfies the BSTP.
  • A non-empty binary tree T satisfies the BSTP if and only if 
    • the left and right subtrees of T both satisfy BSTP, and 
    • all elements in the left subtree of T are less than the root of T, and 
    • the root of T is less than all elements in the right subtree of T.

We can take advantage of this property when looking up for a particular ordered object in the tree.  Instead of scanning the whole tree for the search target, we can compare the search target against the root element and narrow the search to the left subtree or the right subtree if necessary.  So in the worst possible case, the number of comparisons is proportional to the height of the binary tree.  This is a big win if the tree is balanced .  It can be proven that when a tree containing N elements is balanced, its height is at most a constant multiple of logN.  For example, the height of a balanced tree containing 10 6 elements is at most a fixed multiple of 6.  Here is the definition of what a balanced tree is.

Definition of balanced tree

  • An empty tree is balanced .
  • A non-empty tree is balanced if and only if
    •  its subtrees are balanced and
    • the heights of the subtrees differ by a fixed constant or by a fixed constant factor.

A binary tree with  the BST property is called a binary search tree.  It can serve as an efficient way for storage/retrieval of data.  We are lead to the following question: how to create and maintain a binary search tree?  

2. binary search tree insertion

Suppose we start with an empty binary tree T and  a  Comparator that models a total ordering in a given set of objects S.  Then T clearly has the BST property with respect the Comparator ordering of S.  The following algorithm (visitor on binary trees) will allow us to insert elements of S into T and at the same time maintain the BST property for T.  This algorithm also works for binary search tree containing Comparable objects.

Questions & Answers

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
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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Principles of object-oriented programming. OpenStax CNX. May 10, 2013 Download for free at http://legacy.cnx.org/content/col10213/1.37
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