Executive
It is President.
The legislative branch
The power to make war. The power to negotiate or agree to treaties with foreign nations. All these are in Article Four, which includes a prohibition of one state discriminating against the citizens of another state, for instance by making criminal penalties more severe for non-citizens than for citizens.
The House of Representatives has no power with regard to foreign treaties. Treaties with foreign government are negotiated by the President subject to the advice and consent of the Senate only.
the president has the power to create treaties
Recognition of foreign governments is normally shown by the establishment of a consulate, embassy, or special mission to the foreign government in question. These embassies (etc) are established and run by the US Department of State, headed by the Secretary of State (currently Hillary Clinton) and overseen by the President. Therefore, recognition of foreign governments is typically done by the executive branch. Congress has the power to ratify treaties with foreign governments and can pass legislation involving the recognition of or relationships with foreign governments. So sometimes, recognition of foreign governments can be done by the legislative branch. But generally speaking, recognizing foreign governments is a power of the Head of State of a nation, and the Head of State of the United States is the President, so it's an executive branch power.
The president! Article II. Section 2 of the Constitution says: He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur.
Exclusive
exclusive
no
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