Almost every tree company will say they can trim your tree, and most of them will mean it. What separates one estimate from another is what they will actually cut, what they will do with the cuttings, and what happens if something goes wrong while they are working. Those three things are worth comparing side by side before anyone picks up a saw.
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Start with the tree you are actually asking about
Walk the yard with your father and point out what bothers you. Dead limbs, branches that touch the roof or the gutters, and limbs that lean over a driveway or a neighbor's fence are different problems with different fixes. Ask whether the tree needs trimming, removal, or only a clearing of the lines to the house. A company that asks you these questions before quoting is listening. One that quotes the same price without asking has probably not seen the tree.
Compare the written scope
Ask each company to write down exactly which branches they will cut, how high they will work, and whether they will remove any limbs that are already down. Ask whether they will take off dead wood or only shorten live branches. Ask whether they will work near power lines, and if so, whether they will contact the utility before they start. The written scope is what you will check the work against, so it should name the branches rather than say the tree will be trimmed as needed.
Compare the cleanup
Trimming and cleaning up are often two lines on the estimate, and one company may include the haul-away while the other leaves the brush at the curb. Ask where the wood and chips go, whether the lawn and the driveway will be protected from the trucks and the chipper, and whether they will rake up the small debris. If stump grinding is ever part of the conversation, ask whether it is quoted separately, because it usually is.
Compare the proof of coverage
Ask for the certificate of liability insurance and the certificate of workers' compensation coverage, and ask for them in writing, sent to you directly from the insurer or the company. A verbal yes does not protect your father if a limb falls on a fence or a crew member is hurt on the property. Check that the dates on the certificates cover the days of the work, and ask who to call if a problem appears a week later.
How to choose
Put the estimates next to each other and compare the cut list, the cleanup, and the paperwork, in that order. The lowest number may be the company that leaves the most behind. The highest may have written down exactly what they will do. Choose the one you can read back to your father in plain words.
What to do tonight
Go out with a flashlight and look at the branches nearest the roof and the lines. Note the ones that look dead, and write down what you see. That is your starting list for the first call, and your father may have opinions about which limb is his favorite.