The forest can expand into any unoccupied plains or grassland square (even if it has hills). It can't expand into any square you've improved and I've never seen it expand into squares with resources. The same is true for jungle.
As my experience shows it does not expand into desert
Forest expand only in adjacent to forest tiles. but not on diagonal.
Cannot confirm if the chance is greater if you have more than one forest adjacent to the target tile.
The target tile should not have any resource on it - hidden or already revealed.
Improvements on tiles prevent forest from spawning also.
The biggest chance of spawning a forest is on a tile that already had a forest removed before.
Thanks for the responses. Guess I'll go build a mine on that hill after work rather than hoping for the forest on it to expand into the adjacent desert.
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