text:
Hank (textile)
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In the textile industry, a hank is a coiled or wrapped unit of yarn or twine (as opposed to both other objects like thread or rope as well as other forms such as in a ball, cone, bobbin, spool, etc). This is often the best form for use with hand looms, compared to the cone form needed for power looms. Hanks come in varying lengths depending on the type of material and the manufacturer. For instance, a hank of linen is often 300 yards (270 m), and a hank of cotton or silk is 840 yards (770 m).
Hanks, twisted and untwisted. The tie typically used to hold the coil together is visible on the left.
Hanks and skeins of yarn. The three uppermost yarns are in hanks.
While hanks may differ by manufacturer and by product, a skein is usually considered 1/6th of a hank (either by weight or by length). One source identifies a skein of stranded cotton as being 8.25 yards (7.54 m), of tapestry wool as being 10 yards (9.1 m), and crewel wool as being 33 yards (30 m).[1]
In yarns for handcrafts such as knitting or crochet, hanks are not a fixed length but are sold in units by weight, most commonly 50 grams. Depending on the thickness of the strand as well as the inherent density of the material, hanks can range widely in yardage per 50 gram unit; for example, 440 yards for a lace weight mohair, to 60 yards for a chunky weight cotton. Special treatments to the materials that add cost, such as mercerisation or labor-intensive hand-painting of colors, can influence a manufacturer's desired length per unit as well. Knitters and crocheters rewind the hanks into balls or centre-pull skeins prior to use, in order to prevent the yarn from becoming tangled.[2]
Hog casings are sold in "bundles" or "hanks." This unit of measure equals 100 yards (91 m).
References[edit]
Jump up ^ Marie Clayton (10 December 2007). The Ultimate A to Z Companion to 1,001 Needlecraft Terms: Applique, Crochet, Embroidery, Knitting, Quilting, Sewing and More. St. Martin's. p. 143. ISBN 978-0-312-37777-9.
Jump up ^ Ratigan, Dorothy T (2012). Knitting know-how. Durant, Judith, 1955- (1st ed ed.). Cincinnati, Ohio: Krause Publications. p. 11. ISBN 9781440218194. OCLC 748334224. CS1 maint: Extra text (link)
See also[edit]
Spool
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Twelfth Amendment to the United States Constitution
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The Twelfth Amendment (Amendment XII) to the United States Constitution provides the procedure for electing the President and Vice President. It replaced the procedure provided in Article II, Section 1, Clause 3, by which the Electoral College originally functioned. Problems with the original procedure arose in the elections of 1796 and 1800. The Twelfth Amendment refined the process whereby a President and a Vice President are elected by the Electoral College. The amendment was proposed by the Congress on December 9, 1803, and was ratified by the requisite three-fourths of state legislatures on June 15, 1804.
Contents
[hide]
1 Text
2 Background
3 Adoption
3.1 Journey to Congress
3.2 Congressional debate
3.3 Proposal and ratification
4 Electoral College under the Twelfth Amendment
5 Elections since 1804
6 See also
7 Notes
8 References
9 External links
Text[edit]
The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and all persons voted for as Vice-President and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate.
The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted.
The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.[Note 1]
The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.[1]
Background[edit]
Under the original procedure for the Electoral College, as provided in Article II, Section 1, Clause 3, each elector could vote for two persons. The two people chosen by the elector could not both inhabit the same state as that elector. This prohibition was designed to keep electors from voting for two "favorite sons" of their respective states.[2] The person receiving the greatest number of votes, provided that number equaled a majority of the electors, was elected President.
If there were more than one individual who received the same number of votes, and such number equaled a majority of the electors, the House of Representatives would choose one of them to be President. If no individual had a majority, then the House of Representatives would choose from the five individuals with the greatest number of electoral votes. In either case, a majority of state delegations in the House was necessary for a candidate to be chosen to be President.
Selecting the Vice President was a simpler process. Whichever candidate received the second greatest number of votes became Vice President. The Vice President, unlike the President, was not required to receive votes from a majority of the electors. In the event of a tie for second place, the Senate would choose who would be Vice President from those tied, with each Senator casting one vote. It was not specified in the Constitution whether the sitting Vice President could cast a tie-breaking vote for Vice President under the original formula.
Problems emerged as soon as Presidential elections became competitive with George Washington's refusal to seek a third term.
In the 1796 election, John Adams, the Federalist Party presidential candidate, received a majority of the electoral votes. However, the Federalist electors scattered their second votes, resulting in the Democratic-Republican Party presidential candidate, Thomas Jefferson, receiving the second highest number of electoral votes and thus being elected Vice President. It became apparent that if the Vice President was a defeated electoral opponent of the President the two might not work together effectively. By having the President and Vice President elected on a party ticket, this possibility would be eliminated but it raised a different flaw in the arrangements.
On January 6, 1797, Representative William L. Smith of South Carolina responded to this result by presenting a resolution on the floor of the House of Representatives for an amendment to the Constitution requiring each elector to cast one vote for President and another for Vice President.[3] However, no action was taken on his proposal, setting the stage for the deadlocked election of 1800.
The 1800 election exposed a defect in the original formula in that if each member of the Electoral College followed party tickets, there would be a tie between the two candidates from the most popular ticket. The emergence of partisan political activity caused the failure of the original constitutional plan.
The Federalist states in the House of Representatives decided to give their votes to Aaron Burr for President when he and Jefferson tied in the Presidential election of 1800. With help from Alexander Hamilton, the gridlock was finally broken on the thirty-sixth vote and Jefferson was elected President on February 17, 1801.[4] The events in the House, combined with the increasing Democratic-Republican majorities in the House and Senate, prompted the Democratic-Republicans to call for a Constitutional amendment requiring separate votes for President and Vice President in the Electoral College. By doing this, the Democratic-Republicans could dash the Federalists' hopes of having any impact on the Presidential election of 1804.
Adoption[edit]
Journey to Congress[edit]
In March 1801, weeks after the election of 1800 was resolved, two amendments were proposed in the New York state legislature that would form the skeleton of the Twelfth Amendment. Governor John Jay submitted to the state legislature an amendment that would require district election of electors in each state and Jedidiah Peck, a member of the lower house, submitted an amendment to adopt designation for the votes for President and Vice President. The two amendments were not considered until early 1802, as the state legislature took a break for the summer and winter. New York state senator DeWitt Clinton moved for the adoption of the amendment in January 1802 and shortly thereafter won a vacant seat in the U.S. Senate, where he was instrumental in bringing the designation amendment to Congress. The process continued in New York on February 15, when Representative Benjamin Walker of New York proposed the designation and district election amendments to the House. Debate on the amendments began in May and the Republicans wanted to quickly decide on the amendments and go home for the summer, while the Federalists argued that these ideas needed much more time than allowed for in the current session. Federalist Samuel W. Dana of Connecticut wanted an examination of the necessity of a Vice President. The amendment ultimately failed in the New York State Senate, but DeWitt Clinton was able to bring amendment talks to the House.[5]
At this point, debate was ready to begin in Congress, but the Democratic-Republicans decided to wait for the 8th Congress, which would give them a larger majority and thus make the two-thirds requirement for Congress to submit a proposed Constitutional amendment to the states easier to achieve.
Congressional debate[edit]
On its first day, the 8th Congress considered the designation amendment. The first formulation of the amendment had the five highest electoral vote earners on the ballot in the House if no one candidate had a majority of the electoral votes. Republican John Clopton of Virginia, the largest state in the Union, argued that having five names on the list for a contingency election took the power from the people, so he proposed that there be only two names on the list. On October 20, the House appointed a seventeen-member committee (one Representative from each state) to fine-tune the amendment.[6]
The original proposal starting in the New York state legislature was a dual-amendment that, along with designation, put forward the idea of district election of electors that Treasury Secretary Gallatin had supported. Shortly after the committee was formed, Federalist Benjamin Huger attempted to add a provision regarding district elections to the proposed amendment. The committee ignored him and district election disappeared.[7]
The committee then submitted an updated version of the designation amendment to the House on October 23 that changed the number of candidates in a contingency election from five to three and allowed the Senate to choose the Vice President if there were a tie in that race. Small Federalist states disliked the change from five to three, because it made it far less likely that a small-state candidate would make it to a contingency election. Huger and New York Federalist Gaylord Griswold argued that the Constitution was a compromise between large and small states and the method chosen by the Framers is supposed to check the influence of the larger states. Huger even asserted that the Constitution itself was not a union of people, but a union of large and small states in order to justify the original framework for electing the President. Designation, argued Griswold and Huger, would violate the spirit of the Constitution by taking away a check on the power of the large states.[5]
Next up for the Federalists was Seth Hastings of Massachusetts, who submitted the argument that the designation amendment rendered the Vice Presidency useless and advocated for the elimination of the three-fifths clause. John C. Smith asked the inflammatory question of whether or not the proposed amendment was to help the reelection of Jefferson. Speaker Nathaniel Macon called this inappropriate. After Matthew Lyon of Kentucky denounced any reference to the three-fifths clause as mere provocation, the House easily passed the resolution 88-39 on October 28.
By October 28, the Senate had already been discussing the designation amendment. Republican DeWitt Clinton expected that the Senate, with a 24-9 Republican majority would quickly passing the amendment. Federalist Jonathan Dayton proposed that the office of the Vice President should be eliminated and his colleague, Uriah Tracy, seconded it. On the other side, Wilson Cary Nicholas was simply worried that Congress would not submit the amendment in time for the states to ratify it before the 1804 election. Despite Nicholas’ concern, the Senate would not seriously deal with the amendment again until November 23.[8]
Much as it had in the House, debate centered around the number of candidates in a contingency election and the philosophical underpinnings of the Constitution. Again, small Federalist states vehemently argued that three candidates gave too much power to large states to pick presidents. Senator Pierce Butler of South Carolina argued that the issues with the election of 1800 were unlikely to happen again and he would not advocate changing the Constitution simply to stop a Federalist Vice President. John Quincy Adams argued that the change from five to three gave an advantage to the people that violated the federative principle of the Constitution. Rather than have the office of the President balanced between the states and the people, Adams felt designation of President and Vice President would tip that scale in favor of the people.[9]
Federalist Senators argued for retaining the original procedure for the Electoral College. Senator Samuel White of Delaware claimed that the original procedure had not been given "a fair experiment" and criticized the proposed amendment for entrenching the two-party system that had taken over presidential elections.[10]
In response, the Republicans appealed to democratic principles. Samuel Smith of Maryland argued that the presidency ought to be as closely accountable to the people as possible. As such, having three candidates in a contingency election is far better than having five, because it is possible to have the fifth best candidate become President otherwise. Also, designation itself would drastically cut down the number of elections that would reach the house, and the President is then much more likely to be the people’s choice. Another of Smith’s arguments was simply the election of 1800. William Cocke of Tennessee took a different approach when he argued that the entire small state argument of the Federalists was simply out of self-interest.[11]
One last order of business for the amendment was to deal with the possibility that the House would fail to choose a President by March 4. It was the least controversial portion of the Twelfth Amendment and John Taylor proposed that the Vice President would take over as President in that peculiar occurrence, “as in case of the death or other Constitutional disability of the President."[12]
It seemed clear all along that the Republican dominance would render this a no-contest and the Republicans were just waiting for all of their votes to be present, but the Federalists had one last defense. A marathon session of debate, lasting from 11 A.M. to 10 P.M., was the order of the day on December 2, 1803. Most notably, Uriah Tracy of Connecticut argued in a similar vein as Adams when he invoked the federative principle of the Constitution. For Tracy, the original two-vote system was formulated in order to give the small states a chance to elect the Vice President, who would be a check on the President’s powers. In essence, the states balanced the power of the people. However, this only works if you make it partisan, as Georgia (for example) was a Republican small state.[13]
Proposal and ratification[edit]
Twelfth Amendment in the National Archives
The Twelfth Amendment was proposed by the 8th Congress on December 9, 1803, when it was approved by the House of Representatives by vote of 83–42,[14] having been previously passed by the Senate, 22–10, on December 2.[15] The amendment was officially submitted to the states on December 12, 1803, and was ratified by the legislatures of the following states:[16]
North Carolina: December 21, 1803
Maryland: December 24, 1803
Kentucky: December 27, 1803
Ohio: December 30, 1803
Pennsylvania: January 5, 1804
Vermont: January 30, 1804
Virginia: February 3, 1804
New York: February 10, 1804
New Jersey: February 22, 1804
Rhode Island: March 12, 1804
South Carolina: May 15, 1804
Georgia: May 19, 1804
New Hampshire: June 15, 1804[Note 2]
Having been ratified by the requisite three-fourths of the several states, there being 17 states in the Union at the time, the ratification of the Twelfth Amendment was completed. It was subsequently ratified by:
14. Tennessee: July 27, 1804
15. Massachusetts: 1961 (after rejection: February 3, 1804)
The amendment was rejected by:
Delaware: January 18, 1804
Connecticut: May 10, 1804
On September 25, 1804, in a circular letter to the governors of the states, Secretary of State Madison declared the amendment ratified by three-fourths of the states.
Electoral College under the Twelfth Amendment[edit]
While the Twelfth Amendment did not change the composition of the Electoral College or the duties of the electors, it did change the process whereby a President and a Vice President are elected. The new electoral process was first used for the 1804 election. Each presidential election since has been conducted under the terms of the Twelfth Amendment.
The Twelfth Amendment stipulates that each elector must cast distinct votes for President and Vice President, instead of two votes for President. Additionally, electors may not vote for presidential and vice-presidential candidates who both reside in the elector's state—at least one of them must be an inhabitant of another state.
If no candidate for President has a majority of the total votes, the House of Representatives, voting by states and with the same quorum requirements as under the original procedure, chooses the President. The Twelfth Amendment requires the House to choose from the three highest receivers of electoral votes, compared to five under the original procedure.
The Twelfth Amendment requires a person to receive a majority of the electoral votes for Vice President for that person to be elected Vice President by the Electoral College. If no candidate for Vice President has a majority of the total votes, the Senate, with each Senator having one vote, chooses the Vice President. The Twelfth Amendment requires the Senate to choose between the candidates with the "two highest numbers" of electoral votes. If multiple individuals are tied for second place, the Senate may consider all of them, in addition to the individual with the greatest number of votes. The Twelfth Amendment introduced a quorum requirement of two-thirds of the whole number of Senators for the conduct of balloting. Furthermore, the Twelfth Amendment requires the Senate to choose a Vice President by way of the affirmative votes of "a majority of the whole number" of Senators.
To prevent deadlocks from keeping the nation leaderless, the Twelfth Amendment provided that if the House did not choose a President before March 4 (then the first day of a Presidential term), the individual elected Vice President would "act as President, as in the case of the death or other constitutional disability of the President." The Twelfth Amendment did not state for how long the Vice President would act as President or if the House could still choose a President after March 4. Section 3 of the Twentieth Amendment, adopted in 1933, supersedes that provision of the Twelfth Amendment by changing the date upon which a new presidential term commences to January 20, clarifying that the Vice President-elect would only "act as President" if the House has not chosen a President by January 20, and permitting Congress to statutorily provide "who shall then act as President, or the manner in which one who is to act shall be selected" if there is no President-elect or Vice President-elect by January 20. It also clarifies that if there is no President-elect on January 20, whoever acts as President does so until a person "qualified" to occupy the Presidency is elected to be President.
The Twelfth Amendment explicitly states the constitutional requirements as provided for the President also apply to being Vice President. It is unclear whether a two-term president could later serve as Vice President. Some argue that the Twenty-second Amendment and Twelfth Amendment both bar any two-term president from later serving as Vice President as well as from succeeding to the presidency from any point in the United States presidential line of succession.[17] Others contend that the Twelfth Amendment concerns qualification for service, while the Twenty-second Amendment concerns qualifications for election, and thus a former two-term president is still eligible to serve as Vice President.[18] The applicability of this distinction has not been tested, as no former president has ever sought the vice presidency, and thus the courts have never been required to make a judgment regarding the matter. During Hillary Clinton's 2016 candidacy, she jokingly said that she had considered naming Bill Clinton as her Vice President, but was advised it would be unconstitutional.[19]
Elections since 1804[edit]
Certificate for the electoral vote for Rutherford B. Hayes and William A. Wheeler for the State of Louisiana
Starting with the election of 1804, each Presidential election has been conducted under the Twelfth Amendment.
Only once since then has the House of Representatives chosen the President: In 1824, Andrew Jackson received 99 electoral votes, John Quincy Adams (son of John Adams) 84, William H. Crawford 41 and Henry Clay 37. All of the candidates were members of the Democratic-Republican Party (though there were significant political differences among them), and each had fallen short of the 131 votes necessary to win. Because the House could only consider the top three candidates, Clay could not become President. Crawford's poor health following a stroke made his election by the House unlikely. Andrew Jackson expected the House to vote for him, as he had won a plurality of the popular and electoral votes.[Note 3] Instead, the House elected Adams on the first ballot with thirteen states, followed by Jackson with seven and Crawford with four.[20] Clay had endorsed Adams for the Presidency; the endorsement carried additional weight because Clay was the Speaker of the House. When Adams later appointed Clay his Secretary of State, many—particularly Jackson and his supporters—accused the pair of making a "Corrupt Bargain".[21][22] In the less contested election for vice president, John C. Calhoun received 182 electoral votes and was elected outright.
In 1836, the Whig Party nominated different candidates in different regions in the hopes of splintering the electoral vote and denying Martin Van Buren, the Democratic candidate, a majority in the Electoral College, thereby throwing the election into the Whig-controlled House. However, this strategy failed with Van Buren winning majorities of both the popular and electoral vote. In that same election no candidate for Vice President secured a majority in the electoral college as Democratic Vice Presidential nominee Richard Mentor Johnson did not receive the electoral votes of Democratic electors from Virginia, because of his relationship with a former slave. As a result, Johnson received 147 electoral votes, one vote short of a majority; to be followed by Francis Granger with 77, John Tyler with 47 and William Smith with 23. This caused the Senate to choose whether Johnson or Granger would be the new Vice President. Johnson won with 33 votes, with Granger receiving 16.[23]
Since 1836, no major U.S. party has nominated multiple regional presidential or vice presidential candidates in an election. However, since the Civil War there have been two serious attempts by Southern-based parties to run regional candidates in hopes of denying either of the two major candidates an electoral college majority. Both attempts (in 1948 and 1968) failed, but not by much—in both cases a shift in the result of two close states would have forced the respective elections into the House.[24][25]
In 1992, Ross Perot specifically cited his disinclination to forcing an election into the House when, in July of 1992, he temporarily suspended his independent candidacy. When he resumed his candidacy in early October 1992, the issue of that year's election being decided in the House of Representatives never came up significantly in the last month of the campaign.[citation needed]
In modern elections, a running mate is often selected in order to appeal to a different set of voters. A Habitation Clause issue arose during the 2000 presidential election contested by George W. Bush (running-mate Dick Cheney) and Al Gore (running-mate Joe Lieberman), because it was alleged that Bush and Cheney were both inhabitants of Texas and that the Texas electors therefore violated the Twelfth Amendment in casting their ballots for both. Bush's residency was unquestioned, as he was Governor of Texas at the time. Cheney and his wife had moved to Dallas five years earlier when he assumed the role of chief executive at Halliburton. Cheney grew up in Wyoming and had represented it in Congress. A few months before the election, he switched his voter registration and driver's license to Wyoming and put his home in Dallas up for sale. Three Texas voters challenged the election in a federal court in Dallas and then appealed the decision to the United States Court of Appeals for the Fifth Circuit, where it was dismissed.[26]
See also[edit]
Electoral College (United States)
United States presidential eligibility legislation
Notes[edit]
Jump up ^ This sentence is superseded by Section 3 of the Twentieth Amendment
Jump up ^ Ratification was completed on June 15, 1804, when the New Hampshire legislature ratified the amendment. The state's governor, John Taylor Gilman, vetoed the resolution of ratification on June 20, and the act failed to pass again by the two-thirds vote then required by the state constitution. However, Article V of the U.S. Constitution specifies that amendments shall become effective "when ratified by the legislatures of three-fourths of the several States or by conventions in three-fourths thereof" and so it has been generally believed that an approval or veto by a governor is without significance.
Jump up ^ The popular vote in 1824 did not consist of all of the states, since many states chose their electors through their legislatures instead of by a vote of their people.
References[edit]
Jump up ^ "Constitution of the United States: Amendments 11-27". National Archives. Retrieved 2008-02-09.
Jump up ^ The Electoral College – Origin and History Dave Leip's Atlas of U.S. Presidential Elections
Jump up ^ United States Congress (1797). Annals of Congress. 4th Congress, 2nd Session. p. 1824. Retrieved June 26, 2006.
Jump up ^ Wood, Gordon (2009). Empire of Liberty: A History of the Early Republic, 1789-1815. New York: Oxford University Press. p. 285.
^ Jump up to: a b Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. pp. 127–128.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. p. 128.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. pp. 133–136.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. pp. 136–137.
Jump up ^ Alder, Carolyn (March 3, 2016). "A Far Superior Method—the Original Electoral College". Freedom Formula. Retrieved June 4, 2017.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. p. 137.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. p. 139.
Jump up ^ Kuroda, Tadahisa (1999). The Origins of the Twelfth Amendment: The Electoral College in the Early Republic. Westport, CN: Greenwood Press. pp. 140–141.
Jump up ^ "13 Annals of Congress 775 (1805)". Retrieved April 18, 2014.
Jump up ^ "13 Annals of Congress 209 (1805)". Retrieved April 18, 2014.
Jump up ^ "THE CONSTITUTION of the UNITED STATES OF AMERICA: ANALYSIS AND INTERPRETATION, Centennial Edition, INTERIM EDITION: ANALYSIS OF CASES DECIDED BY THE SUPREME COURT OF THE UNITED STATES TO JUNE 26, 2013" (PDF). Washington, D.C.: U.S. Government Printing Office. 2013. p. 28. Retrieved April 13, 2014.
Jump up ^ Matthew J. Franck (July 31, 2007). "Constitutional Sleight of Hand". National Review. Retrieved April 20, 2014.
Jump up ^ Scott E. Gant; Bruce G. Peabody (2006-06-13). "How to bring back Bill". The Christian Science Monitor. Retrieved 2008-06-12.
Jump up ^ 1339 GMT (2039 HKT) September 15, 2015 (September 15, 2015). "Hillary Clinton: Bill as VP has 'crossed her mind'". Edition.cnn.com. Retrieved October 29, 2015.
Jump up ^ Andrew Costly. "BRIA 8 4 a The Election of 1824-25: When the House Chose the President - Constitutional Rights Foundation".
Jump up ^ McNamara, Robert. "The Election of 1824 Was Decided in the House of Representatives". About.com. Retrieved 2009-07-03.
Jump up ^ Parsons, Lynn Hudson (2009). The Birth of Modern Politics: Andrew Jackson, John Quincy Adams, and the Election of 1828. Oxford University Press. p. 106. ISBN 978-0-19-975424-3.
Jump up ^ "VP Richard Mentor Johnson". 30 May 2014.
Jump up ^ "Student Activity: Presidential Campaign of 1948". Harry S. Truman Library & Museum. Retrieved March 19, 2016.
Jump up ^ "American Experience: George Wallace - 1968 Campaign". PBS. 2000. Retrieved March 19, 2016.
Jump up ^ Bravin, Jess. Obscure Texas Case Offers Peek Into Role Of Court Nominee, The Wall Street Journal, Oct. 7, 2005.
External links[edit]
Constitution of the United States, via Wikisource
Twelfth Amendment, via National Archives
CRS Annotated Constitution: Twelfth Amendment, via Cornell Law School
Ellis, E. S. (1903). Thomas Jefferson: A Character Sketch., via Project Gutenberg
U. S. Electoral College, via Office of the Federal Register
Hawley, Joshua. "The Transformative Twelfth Amendment". University of Missouri School of Law.
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Find My Friends
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Find My Friends
Find My Friends on the iPad
Developer(s)
Apple Inc.
Initial release
October 12, 2011 (2011-10-12)
Stable release
4.0.1[1] / October 17, 2014; 2 years ago (2014-10-17)[1]
Development status
Active
Operating system
iOS 8 and later
Size
6.1 MB
Type
Location aware mobile application
License
Freeware
Website
www.apple.com/it/icloud/features/find-my.html
Find My Friends (also known as "Find Friends" on the SpringBoard) is an application and service for iOS developed by Apple Inc. It was released on October 12, 2011 to the general public. As of the release of iOS 9, the app is now bundled with the operating system. If a user is running iOS 8, the app is available for download from the App Store free of charge.
Contents
[hide]
1 History
1.1 Version history
2 Features
3 See also
4 References
History[edit]
Find My Friends was announced during the "Let's Talk iPhone" keynote event on October 4, 2011.[2] Find My Friends was released several hours earlier than the actual release of iOS 5.[3]
Version history[edit]
Version Number
Date released
Changes
1.0
October 11, 2011 (2011-10-11)
Initial release
1.1
March 7, 2012 (2012-03-07)
Support for Retina display devices
Bug fixes and enhancements
2.0
September 19, 2012 (2012-09-19)
Location-based alerts (requires iOS 6 or later)
Friend Suggestions and Favourites
2.0.1
October 29, 2012 (2012-10-29)
Temporary friends are listed as Followers
2.1
March 28, 2013 (2013-03-28)
New design and layout
Location-based notifications (iOS 6.1 or later)
2.1.1
August 22, 2013 (2013-08-22)
Bug fixes and stability improvements
3.0
November 19, 2013 (2013-11-19)
New design for iOS 7 devices
4.0
September 17, 2014 (2014-09-17)
Support for iOS 8
Allows users to move their friends list to their iCloud if needed.
4.0.1
October 16, 2014 (2014-10-16)
Bug fixes and stability improvements
Features[edit]
With Find My Friends, users can follow and track people with their iOS device(s).[4] Users can also share their location with the people they choose. Location is determined using GPS in the iOS device when Location Services are turned on. Modified software can allow notifications appear to when a user requests another user to see where they are[5] or is turned on through the message application. It can also be accepted through Facebook sharing options. The feature can be turned on and off at any time.[6] Like many iOS applications that use Location Services, parental controls are available. Find My Friends synchronizes with other applications such as Maps and Contacts. The application is supported on the iPhone, iPod Touch, and iPad.[1] A friend's location can be viewed in OS X 10.10 as well, by clicking "Details" in the top right corner of the Messages App.
See also[edit]
RoomFeed
Find My iPhone
Google Latitude
Life360
iPhone 4S
iCloud
References[edit]
^ Jump up to: a b c "App Store - Find My Friends". Application Store. Apple Inc.
Jump up ^ ZL Studios (October 4, 2011). "Apple Announces Find My Friends iOS 5 App « ZL Studios". Zlstudios.net. Retrieved October 13, 2011.
Jump up ^ Christian Zibreg. "Find My Friends app goes live ahead of iOS 5 release | 9to5Mac | Apple Intelligence". 9to5Mac. Retrieved October 13, 2011.
Jump up ^ "iCloud - Find your friends and your missing Apple device". Apple. Retrieved October 13, 2011.
Jump up ^ "How to Know if Your iPhone Is Being Tracked via Find My Friends - Cydia Geeks". Cydia Geeks. 2017-01-20. Retrieved 2017-01-20.
Jump up ^ Previous post Next post (January 4, 2009). "Apple Stealth Launches ‘Find My Friends’ and ‘Airport Utility’ Apps | Gadget Lab". Wired.com. Retrieved October 13, 2011.
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Binary search tree
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Binary search tree
Type
tree
Invented
1960
Invented by
P.F. Windley, A.D. Booth, A.J.T. Colin, and T.N. Hibbard
Time complexity in big O notation
Algorithm
Average Worst Case
Space
O(n) O(n)
Search
O(log n) O(n)
Insert
O(log n) O(n)
Delete
O(log n) O(n)
A binary search tree of size 9 and depth 3, with 8 at the root. The leaves are not drawn.
In computer science, binary search trees (BST), sometimes called ordered or sorted binary trees, are a particular type of container: data structures that store "items" (such as numbers, names etc.) in memory. They allow fast lookup, addition and removal of items, and can be used to implement either dynamic sets of items, or lookup tables that allow finding an item by its key (e.g., finding the phone number of a person by name).
Binary search trees keep their keys in sorted order, so that lookup and other operations can use the principle of binary search: when looking for a key in a tree (or a place to insert a new key), they traverse the tree from root to leaf, making comparisons to keys stored in the nodes of the tree and deciding, based on the comparison, to continue searching in the left or right subtrees. On average, this means that each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the logarithm of the number of items stored in the tree. This is much better than the linear time required to find items by key in an (unsorted) array, but slower than the corresponding operations on hash tables.
Several variants of the binary search tree have been studied in computer science; this article deals primarily with the basic type, making references to more advanced types when appropriate.
Contents
[hide]
1 Definition
1.1 Order relation
2 Operations
2.1 Searching
2.2 Insertion
2.3 Deletion
2.4 Traversal
2.5 Verification
3 Examples of applications
3.1 Sort
3.2 Priority queue operations
4 Types
4.1 Performance comparisons
4.2 Optimal binary search trees
5 See also
6 Notes
7 References
8 Further reading
9 External links
Definition[edit]
A binary search tree is a rooted binary tree, whose internal nodes each store a key (and optionally, an associated value) and each have two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search tree property, which states that the key in each node must be greater than or equal to any key stored in the left sub-tree, and less than or equal to any key stored in the right sub-tree.[1]:287 (The leaves (final nodes) of the tree contain no key and have no structure to distinguish them from one another. Leaves are commonly represented by a special leaf or nil symbol, a NULL pointer, etc.)
Generally, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records.
The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient; they are also easy to code.
Binary search trees are a fundamental data structure used to construct more abstract data structures such as sets, multisets, and associative arrays. Some of their disadvantages are as follows:
The shape of the binary search tree depends entirely on the order of insertions and deletions, and can become degenerate.
When inserting or searching for an element in a binary search tree, the key of each visited node has to be compared with the key of the element to be inserted or found.
The keys in the binary search tree may be long and the run time may increase.
After a long intermixed sequence of random insertion and deletion, the expected height of the tree approaches square root of the number of keys, √n, which grows much faster than log n.
Order relation[edit]
Binary search requires an order relation by which every element (item) can be compared with every other element in the sense of a total preorder. The part of the element which effectively takes place in the comparison is called its key. Whether duplicates, i.e. different elements with same key, shall be allowed in the tree or not, does not depend on the order relation, but on the application only.
In the context of binary search trees a total preorder is realized most flexibly by means of a three-way comparison subroutine.
Operations[edit]
Binary search trees support three main operations: insertion of elements, deletion of elements, and lookup (checking whether a key is present).
Searching[edit]
Searching a binary search tree for a specific key can be programmed recursively or iteratively.
We begin by examining the root node. If the tree is null, the key we are searching for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and we return the node. If the key is less than that of the root, we search the left subtree. Similarly, if the key is greater than that of the root, we search the right subtree. This process is repeated until the key is found or the remaining subtree is null. If the searched key is not found after a null subtree is reached, then the key is not present in the tree. This is easily expressed as a recursive algorithm (implemented in Python):
1 def search_recursively(key, node): 2 if node is None or node.key == key: 3 return node 4 if key < node.key: 5 return search_recursively(key, node.left) 6 # key > node.key 7 return search_recursively(key, node.right)
The same algorithm can be implemented iteratively:
1 def search_iteratively(key, node): 2 current_node = node 3 while current_node is not None: 4 if key == current_node.key: 5 return current_node 6 elif key < current_node.key: 7 current_node = current_node.left 8 else: # key > current_node.key: 9 current_node = current_node.right 10 return current_node
These two examples rely on the order relation being a total order.
If the order relation is only a total preorder a reasonable extension of the functionality is the following: also in case of equality search down to the leaves in a direction specifiable by the user. A binary tree sort equipped with such a comparison function becomes stable.
Because in the worst case this algorithm must search from the root of the tree to the leaf farthest from the root, the search operation takes time proportional to the tree's height (see tree terminology). On average, binary search trees with n nodes have O(log n) height.[note 1] However, in the worst case, binary search trees can have O(n) height, when the unbalanced tree resembles a linked list (degenerate tree).
Insertion[edit]
Insertion begins as a search would begin; if the key is not equal to that of the root, we search the left or right subtrees as before. Eventually, we will reach an external node and add the new key-value pair (here encoded as a record 'newNode') as its right or left child, depending on the node's key. In other words, we examine the root and recursively insert the new node to the left subtree if its key is less than that of the root, or the right subtree if its key is greater than or equal to the root.
Here's how a typical binary search tree insertion might be performed in a binary tree in C++:
Node* insert(Node* root, int key, int value) { if (!root) root = new Node(key, value); else if (key < root->key) root->left = insert(root->left, key, value); else // key >= root->key root->right = insert(root->right, key, value); return root; }
The above destructive procedural variant modifies the tree in place. It uses only constant heap space (and the iterative version uses constant stack space as well), but the prior version of the tree is lost. Alternatively, as in the following Python example, we can reconstruct all ancestors of the inserted node; any reference to the original tree root remains valid, making the tree a persistent data structure:
 def binary_tree_insert(node, key, value): if node is None: return NodeTree(None, key, value, None) if key == node.key: return NodeTree(node.left, key, value, node.right) if key < node.key: return NodeTree(binary_tree_insert(node.left, key, value), node.key, node.value, node.right) else: return NodeTree(node.left, node.key, node.value, binary_tree_insert(node.right, key, value))
The part that is rebuilt uses O(log n) space in the average case and O(n) in the worst case.
In either version, this operation requires time proportional to the height of the tree in the worst case, which is O(log n) time in the average case over all trees, but O(n) time in the worst case.
Another way to explain insertion is that in order to insert a new node in the tree, its key is first compared with that of the root. If its key is less than the root's, it is then compared with the key of the root's left child. If its key is greater, it is compared with the root's right child. This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left child, depending on its key: if the key is less than the leaf's key, then it is inserted as the leaf's left child, otherwise as the leaf's right child.
There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure.
Deletion[edit]
When removing a node from a binary search tree it is mandatory to maintain the in-order sequence of the nodes. There are many possibilities to do this. However, the following method which has been proposed by T. Hibbard in 1962[2] guarantees that the heights of the subject subtrees are changed by at most one. There are three possible cases to consider:
Deleting a node with no children: simply remove the node from the tree.
Deleting a node with one child: remove the node and replace it with its child.
Deleting a node with two children: call the node to be deleted D. Do not delete D. Instead, choose either its in-order predecessor node or its in-order successor node as replacement node E (s. figure). Copy the user values of E to D.[note 2] If E does not have a child simply remove E from its previous parent G. If E has a child, say F, it is a right child. Replace E with F at E's parent.
Deleting a node with two children from a binary search tree. First the leftmost node in the right subtree, the in-order successor E, is identified. Its value is copied into the node D being deleted. The in-order successor can then be easily deleted because it has at most one child. The same method works symmetrically using the in-order predecessor C.
In all cases, when D happens to be the root, make the replacement node root again.
Broadly speaking, nodes with children are harder to delete. As with all binary trees, a node's in-order successor is its right subtree's left-most child, and a node's in-order predecessor is the left subtree's right-most child. In either case, this node will have only one or no child at all. Delete it according to one of the two simpler cases above.
Consistently using the in-order successor or the in-order predecessor for every instance of the two-child case can lead to an unbalanced tree, so some implementations select one or the other at different times.
Runtime analysis: Although this operation does not always traverse the tree down to a leaf, this is always a possibility; thus in the worst case it requires time proportional to the height of the tree. It does not require more even when the node has two children, since it still follows a single path and does not visit any node twice.
def find_min(self): # Gets minimum node in a subtree current_node = self while current_node.left_child: current_node = current_node.left_child return current_node def replace_node_in_parent(self, new_value=None): if self.parent: if self == self.parent.left_child: self.parent.left_child = new_value else: self.parent.right_child = new_value if new_value: new_value.parent = self.parent def binary_tree_delete(self, key): if key < self.key: self.left_child.binary_tree_delete(key) elif key > self.key: self.right_child.binary_tree_delete(key) else: # delete the key here if self.left_child and self.right_child: # if both children are present successor = self.right_child.find_min() self.key = successor.key successor.binary_tree_delete(successor.key) elif self.left_child: # if the node has only a *left* child self.replace_node_in_parent(self.left_child) elif self.right_child: # if the node has only a *right* child self.replace_node_in_parent(self.right_child) else: # this node has no children self.replace_node_in_parent(None)
Traversal[edit]
Main article: Tree 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. As with all binary trees, one may conduct a pre-order traversal or a post-order traversal, but neither are likely to be useful for binary search trees. An in-order traversal of a binary search tree will always result in a sorted list of node items (numbers, strings or other comparable items).
The code for in-order traversal in Python is given below. It will call callback (some function the programmer wishes to call on the node's value, such as printing to the screen) for every node in the tree.
def traverse_binary_tree(node, callback): if node is None: return traverse_binary_tree(node.leftChild, callback) callback(node.value) traverse_binary_tree(node.rightChild, callback)
Traversal requires O(n) time, since it must visit every node. This algorithm is also O(n), so it is asymptotically optimal.
Traversal can also be implemented iteratively. For certain applications, e.g. greater equal search, approximative search, an operation for single step (iterative) traversal can be very useful. This is, of course, implemented without the callback construct and takes O(1) on average and O(log n) in the worst case.
Verification[edit]
Sometimes we already have a binary tree, and we need to determine whether it is a BST. This problem has a simple recursive solution.
The BST property—every node on the right subtree has to be larger than the current node and every node on the left subtree has to be smaller than (or equal to - should not be the case as only unique values should be in the tree - this also poses the question as to if such nodes should be left or right of this parent) the current node—is the key to figuring out whether a tree is a BST or not. The greedy algorithm – simply traverse the tree, at every node check whether the node contains a value larger than the value at the left child and smaller than the value on the right child – does not work for all cases. Consider the following tree:
     20
    /  \
  10    30
       /  \
      5    40

In the tree above, each node meets the condition that the node contains a value larger than its left child and smaller than its right child hold, and yet it is not a BST: the value 5 is on the right subtree of the node containing 20, a violation of the BST property.
Instead of making a decision based solely on the values of a node and its children, we also need information flowing down from the parent as well. In the case of the tree above, if we could remember about the node containing the value 20, we would see that the node with value 5 is violating the BST property contract.
So the condition we need to check at each node is:
if the node is the left child of its parent, then it must be smaller than (or equal to) the parent and it must pass down the value from its parent to its right subtree to make sure none of the nodes in that subtree is greater than the parent
if the node is the right child of its parent, then it must be larger than the parent and it must pass down the value from its parent to its left subtree to make sure none of the nodes in that subtree is lesser than the parent.
A recursive solution in C can explain this further:
struct TreeNode { int key; int value; struct TreeNode *left; struct TreeNode *right; }; bool isBST(struct TreeNode *node, int minKey, int maxKey) { if(node == NULL) return true; if(node->key < minKey || node->key > maxKey) return false; return isBST(node->left, minKey, node->key-1) && isBST(node->right, node->key+1, maxKey); }
node->key+1 and node->key-1 are done to allow only distinct elements in BST.
If we want same elements to also be present, then we can use only node->key in both places.

The initial call to this function can be something like this:
if(isBST(root, INT_MIN, INT_MAX)) { puts("This is a BST."); } else { puts("This is NOT a BST!"); }
Essentially we keep creating a valid range (starting from [MIN_VALUE, MAX_VALUE]) and keep shrinking it down for each node as we go down recursively.
As pointed out in section #Traversal, an in-order traversal of a binary search tree returns the nodes sorted. Thus we only need to keep the last visited node while traversing the tree and check whether its key is smaller (or smaller/equal, if duplicates are to be allowed in the tree) compared to the current key.
Examples of applications[edit]
Some examples shall illustrate the use of above basic building blocks.
Sort[edit]
Main article: Tree sort
A binary search tree can be used to implement a simple sorting algorithm. Similar 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.
The worst-case time of build_binary_tree is O(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 (1 (2 (3 (4 (5))))).
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(n log n), which is asymptotically optimal for a comparison sort. In practice, the 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 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.
Priority queue operations[edit]
Binary search trees can serve as priority queues: structures that allow insertion of arbitrary key as well as lookup and deletion of the minimum (or maximum) key. Insertion works as previously explained. Find-min walks the tree, following left pointers as far as it can without hitting a leaf:
// Precondition: T is not a leaf function find-min(T): while hasLeft(T): T ? left(T) return key(T)

Find-max is analogous: follow right pointers as far as possible. Delete-min (max) can simply look up the minimum (maximum), then delete it. This way, insertion and deletion both take logarithmic time, just as they do in a binary heap, but unlike a binary heap and most other priority queue implementations, a single tree can support all of find-min, find-max, delete-min and delete-max at the same time, making binary search trees suitable as double-ended priority queues.[3]:156
Types[edit]
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 (randomly chosen) priority and the parent node has higher priority than its children. Tango trees are trees optimized for fast searches. T-trees are binary search trees optimized to reduce storage space overhead, widely used for in-memory databases
A degenerate tree is a tree where for each parent node, there is only one associated child node. It is unbalanced and, in the worst case, performance degrades to that of a linked list. If your add node function does not handle re-balancing, then you can easily construct a degenerate tree by feeding it with data that is already sorted. What this means is that in a performance measurement, the tree will essentially behave like a linked list data structure.
Performance comparisons[edit]
D. A. Heger (2004)[4] 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 variations.
Optimal binary search trees[edit]
Main article: Optimal binary search tree
Tree rotations are very common internal operations in binary trees to keep perfect, or near-to-perfect, internal balance in the tree.
If we do not plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct[5] 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.
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 corpora, 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 store data elements only 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).
See also[edit]
Binary search algorithm
Search tree
Self-balancing binary search tree
AVL tree
Red–black tree
Randomized binary search tree
Tango tree
Notes[edit]
Jump up ^ The notion of an average BST is made precise as follows. Let a random BST be one built using only insertions out of a sequence of unique elements in random order (all permutations equally likely); then the expected height of the tree is O(log n). If deletions are allowed as well as insertions, "little is known about the average height of a binary search tree".[1]:300
Jump up ^ Of course, a generic software package has to work the other way around: It has to leave the user data untouched and to furnish E with all the BST links to and from D.
References[edit]
^ Jump up to: a b Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. ISBN 0-262-03384-4.
Jump up ^ s. Robert Sedgewick, Kevin Wayne: Algorithms Fourth Edition. Pearson Education, 2011, ISBN 978-0-321-57351-3, p. 410.
Jump up ^ Mehlhorn, Kurt; Sanders, Peter (2008). Algorithms and Data Structures: The Basic Toolbox (PDF). Springer.
Jump up ^ Heger, Dominique A. (2004), "A Disquisition on The Performance Behavior of Binary Search Tree Data Structures" (PDF), European Journal for the Informatics Professional, 5 (5): 67–75
Jump up ^ Gonnet, Gaston. "Optimal Binary Search Trees". Scientific Computation. ETH Zürich. Archived from the original on 12 October 2014. Retrieved 1 December 2013.
Further reading[edit]
This article incorporates public domain material from the NIST document: Black, Paul E. "Binary Search Tree". Dictionary of Algorithms and Data Structures.
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "12: Binary search trees, 15.5: Optimal binary search trees". Introduction to Algorithms (2nd ed.). MIT Press & McGraw-Hill. pp. 253–272, 356–363. ISBN 0-262-03293-7.
Jarc, Duane J. (3 December 2005). "Binary Tree Traversals". Interactive Data Structure Visualizations. University of Maryland.
Knuth, Donald (1997). "6.2.2: Binary Tree Searching". The Art of Computer Programming. 3: "Sorting and Searching" (3rd ed.). Addison-Wesley. pp. 426–458. ISBN 0-201-89685-0.
Long, Sean. "Binary Search Tree" (PPT). Data Structures and Algorithms Visualization-A PowerPoint Slides Based Approach. SUNY Oneonta.
Parlante, Nick (2001). "Binary Trees". CS Education Library. Stanford University.
External links[edit]
Literate implementations of binary search trees in various languages on LiteratePrograms
Binary Tree Visualizer (JavaScript animation of various BT-based data structures)
Kovac, Kubo. "Binary Search Trees" (Java applet). Korešponden?ný seminár z programovania.
Madru, Justin (18 August 2009). "Binary Search Tree". JDServer. Archived from the original on 28 March 2010. C++ implementation.
Binary Search Tree Example in Python
"References to Pointers (C++)". MSDN. Microsoft. 2005. Gives an example binary tree implementation.
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