As a US citizen living in Europe, must I use a trust as the owner of an N-reg aircraft?
As a US citizen you can own an N-reg aircraft directly, regardless of where you live. The regulations on aircraft registration are in 14 CFR Part 47. They say who can register an N-reg aircraft, but they don't say anything about where that person must be. 47.3(a)(1) simply says:

§47.3   Registration required.
(a) An aircraft may be registered under 49 U.S.C. 44103 only when the
aircraft is not registered under the laws of a foreign country and is—
(1) Owned by a citizen of the United States;

The regulations are summarized in a more readable way on this FAA site. A trust allows a non-US citizen (or non-resident alien) to own an N-reg aircraft provided they have no more than 25% of the voting rights in the trust, per 47.7(c)(2)(iii).