It would be perfectly possible for the forest to have the only sufficient concentration of oxygen on the planet - if the forest was at the bottom of the deepest canyon or trench on the planet, and if the other regions of the planet had a little less oxygen than could support human life.
And possibly the forest will eventually - which might be many millions of years - provide enough surplus oxygen for higher elevations of the planet to have sufficient oxygen.
The lower the elevation the higher the pressure of the atmosphere. But a habitable planet would have to have large bodies or water, so it is actually "the lower the elevation above local water level, the higher the atmosphereic pressure". The higher the elevation, the lower the density of the atmosphere.
On Earth, people who go to much higher elevations than they are used to get altitude sickness, finding it hard to breathe the much thinner air. (And people who go to much lower altitudes than they are used to can suffer reverse altitude sickness trying to breathe the thicker air.) Above about 8,000 meters altitude the air is so thin that mountaineers have to used bottled oxygen, and many still suffer serious altitude sickness in that "death zone".
So you want a planet with an atmosphere thin enough that it has "death zone" thin air all over the surface, except in one very deep region where the air is thick enough.
So you want to check Habitable Planets for Man, Stephen H. Dole, 1964, for the atmospheric requirements of humans.
Dole discusses atmospheric requirements on pages 13 to 19, and in table 4 on page 21.
The oxygen requirement is oxygen pressure between 60 and 400 millimeters of mercury.
So possibly there is a low, low, lowland region with an inland sea with many oxygen producing plants, with a forest of oxygen producing trees surrounding the inland sea, and an atmomsophere dense enough to have about 70 mm of mercury of oxygen at the level of the inland sea and the forest. And most of the planet is high enough above that level to have half the atmospheric pressure of all atmospheric gases, making the atmospheric pressure of oxygen about 35 mm of mercury. And there are tall mountains and high plateaus on the planet where the atmosphere is half an dense as on most of the planet. Thus in those high places the pressure of oxygen would be only about 17.5 mm of mercury.
There may be oxygen producing photosynthasizing plants over most of the planet's surface, but the oxygen they produce doesn't stay local but diffuses all over the planet's surface, so there is equal oxygen whereever the altitude is the same. And only the depression with your forest has a low enough altitude to have concentrated enough oxygen in the atmosphere. Everywhere else on the planet the oxygen pressure is about as low as in the "death zone" on Earth, except that the air is really thin in the high mountains and plateaus.
There was a somewhat similar question asked a while ago.
Walking to space
fortunately you do not need as strong a pressure differential on your world.