Backgrounds aren't built to provide mechanical combat features
Should I simply give him advantage on the attack roll, ignore fall
damage, and call it a day?
If you did that, your PC is doing something like "Slow Fall" which is a class feature (4th level) of a Monk and, when you offer advantage, is getting part of the Assassin (Rogue) archetype from 3rd level. (Assassinate, p. 97, PHB)
In short, no, you should not do that unless you are offering the other PC's a similar 'free' class feature. But, as you are in home brew territory, there are a number of things that you can do.
"It's a part of his background"
Review the backgrounds in the PHB. How many of them provide a tactical mechanic? Spoiler
none of them do. :-)
You can customize a background, but be careful with mechanical combat features if you do.
To customize a Background, you can replace one feature with any other
one, choose any two Skills, and choose a total of two Tool
Proficiencies or Languages from the sample Backgrounds. You can
either use the Equipment package from your Background or spend coin on
gear. Finally, choose two personality traits, one ideal, one bond,
and one flaw. If you can’t find a feature that matches your desired
Background, work with your DM to create one. (PHB p. 125)
Background features include: free/working ship passage(Sailor), guild membership(guild artisan), criminal contacts(Criminal), false identity(Charlatan). Travel Movement bonus (Marine, Ghosts of Saltmarsh)
None of these includes a combat mechanic.
Homebrew of a feat looks like your best way forward here
Take a look at the combat feats in Chapter 6 and tailor one that fits.
- Homebrew feat examples; outline;
- "Tree Fighter"; (1) reduces falling damage by half when making such an attack, (2) granting a bonus on initiative, and (3) perhaps one other benefit like advantage on Stealth checks in the trees ...). This is similar to the feat Skulker, but different.
- A feat that more or less replicates the Monk Slow Fall class feature and perhaps advantage on Stealth checks
Homebrew an archetype/sub class
This is a bit more work, but you can rebuild a Rogue/Fighter/Barbarian/Ranger sub-class from the bottom up to fit this character concept. A couple of examples are here, and the entire series of unearthed Arcana articles include class and sub-class modifications. I'd recommend that you and your player work together on such a creative process. (My nephew built a "Water" Domain cleric with our DM a couple of years ago. It worked out well).
RAF/Big picture question: the campaign - how much is in forests?
For how long is "fighting in trees" a part of your campaign? If most of your campaign is out of doors and in forests, fitting this "tree fighter" bit becomes important for as long as your campaign lasts: to level 6, to level 10, to level, 15, etc.
If, on the other hand, this situation comes up only now and again in play, but most of your campaign is in cities and dungeons and at sea, then substituting in a modified class feature from the background may fit. For example, for this narrow application, reducing this character's falling damage by 1d6 if they are attacking out of trees like that is an occasional benefit, and it doesn't negate the damage usually taken from such an attack. In certain situations this character has an edge, but in most others they are a (whatever class they are).
That last approach is based on "Rules as Fun" more than "Rules as Written" as an approach, but it is a way to proceed.
RAF. Regardless of what’s on the page or what the designers intended,
D&D is meant to be fun, and the DM is the ringmaster at each game
table. The best DMs shape the game on the fly to bring the most
delight to their players. Such DMs aim for RAF, “rules as fun.” We
expect DMs to depart from the rules when running a particular campaign
or when seeking the greatest happiness for a certain group of players.