But I can't see a way to replace the trees. Will they grow back? If so, how fast? Or is there a finite supply of trees/logs?
You can't manually replace wild trees you've cut down. However, they will grow back over time. The placement of new trees is randomly generated so they won't grow back in exactly the same places (or necessarily the same distribution). Trees won't grow back within a certain radius of buildings, either, so don't build near where you want wild trees to grow.
The rate of wild tree growth is slow; it can take several in-game years for a tree to grow to a "full" tree. Luckily, however, most trees can be chopped down again before they are fully-grown. Keep in mind that many trees also allow the collection of fruit, nuts, or tannin, so it's best to avoid chopping those down if you can.
You can also get logs from traders when they arrive in your settlement, if they happen to offer them (each trader's offering is randomized based on your settlement status/progress)
(There is a parameter in your .sav file that controls tree growth speed, I believe, but I don't know exactly where it is. I've only looked at .sav files for Dawn of Man once. If you're bothering with editing the saved game file, though, you might as well just increase the number of logs in your inventory)