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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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