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CONSTITUTION
There were no branches when the Constitution was written. Nor was there a USA. The Constitution is part of the creation of the USA, the branches did not exist at the until the time it was written.
There was no government in place until the Constitution was ratified.
The official that the Texas Constitution specified should be elected is the governor. A governor is an individual public official with the power to govern the executive branch of a non sovereign level of government.
The Constitution was written during the Constitutional Conventional and was originally meant to revise the weak Articles of Confederation. However, what the delegates drew up ended up replacing the Articles of Confederation entirely with a republic with three branches of government: a strong executive branch (the president), the legislative branch (Congress), and the judicial branch (the Supreme Court).
I'm assuming you are referring to the New Hampshire Constitution of 1776 (written four months before the Declaration of Independence). The chief differences between the 1776 Constitution and the US Constitution are a lack of bill of rights, and no executive branch. The two branches were a house of representatives, and a council. The council was originally appointed with the intention of a popular vote if a universal constitution wasn't written, because ultimately this constitution seemed to be written as an emergency constitution until a multi-colony constitution was created.
The Constitution as originally written would have contridicted the statement "All men are created equal." The legislative branch was to be based on population, but "excluding Indians not taxed, three fifths of all other Persons" which was a compromise around the issue of slavery.
The head of the executive branch is elected. He is the president.
Article 2 describes the Executive Branch. It provides the requirements for being president. It also lists the various duties and responsibilities, as well as how the President is to be elected.
Elected
The Judicial Branch is nominated by the Executive Branch, with the concurrance of the Legislative Branch. The Executive Branch are appointed by the President once he has been elected. The Legislative Branch is elected by the people.
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