Question

How many territories were gained by war? how many by treaty? how many by purchase?

Answer 1

See below

Explanation:

This is a difficult question as some of the gains seem of little consequence. But the list is long.

This has a listing of the Lend Lease territorial gains:

https://en.wikipedia.org/wiki/Destroyers_for_Bases_Agreement

The U.S. gained bases in the Pacific that were formerly Japanese territories. Some of which were former German territories from before World War 1. The Marianas, Carolines, Marshall Islands and Palau groups were part of these.

https://en.wikipedia.org/wiki/South_Pacific_Mandate

Here a list of U.S. territorial gains since inception.

https://en.wikipedia.org/wiki/United_States_territorial_acquisitions