Can the executor of a trust evict the beneficiary of the trust in 30 days for lack of lease agreement when the home has been the sole residence of the beneficiary

The answer is "maybe."
The trustee (what you call the executor of the trust) has the legal authority to evict anyone who does not have a written lease on a months' notice; when there is no written lease, you are a month-to-month tenant on an oral lease, which means that your tenancy can be terminated with one month's notice.
(Or if you are not a rent-paying tenant but have been living there without paying rent, you are a guest; and a guest can be asked to leave at any time by those who control the property.)
So, legally, the trustee has the authority to remove you. However, the trustee also has a "fiduciary duty" to the beneficiares--a duty to act in their interest. If you are a beneficiary, the trustee cannot lightly remove you--he should only do so if it is at least reasonably in your interest (such as if the home is too large for you, too expensive to keep up, or if you need the proceeds from renting or selling it, which is easier to do if you are out). If you feel that being removed is not in your interest, as the beneficiary of the trust, you can bring a legal action in court to have a judge determine if the trustee is acting in accordance with his duty or is violating it (and if so, should not be allowed to do it).