After some testing yes the core works still if moved by a gravity gun(have not tried portal)
Tree of time radius is a chunk last I checked on the twilight forest wiki
mining tree seems to pull from two to one chunks around it
Well, what I was questioning was not if they can be moved by the Gravity Gun/Portal Gun, but more along the lines if the tree's core will continue to function without the rest of the tree (e.g. if the rest of the tree was chopped down, and leaving only the Core block).
And as for the what you said on the radius, what do you mean by "a chunk"? Do you mean as in a actual 16 block radius around the Tree, or more like it only affects things within the same chunk (Like how a Spot Loader only loads things in the same chunk)?
I did notice that the Tree of Transformation has a effective area of 16 blocks on the x/z axis and a indefinite radius along the y axis (seeing as it is changing the Biome and biomes do not change along the y axis by default). So if the Tree of Time and Tree of Sorting shares this trait with the Tree of Transformation, I could have effective just answer my own question.
As for the Miner's Tree, saying that it pulls something from about a "chunk" or 2 would implies that it would have a radius of around 16~32 blocks. So far, I have noticed that it only pulls ores from about 32 blocks below, much like the Ore Magnet, but I am also wondering of it X/Z Radius (as in it's radius along the x or z axis). This seems to be slightly different then the 32 block for the Y axis, seeing as it pulled some
Gravel Iron Ore (from Tinkers Construct) that I know was about 40 blocks away up higher (though the Gravel Ores don't seem to actually move along the x/z axis like other ores do).
Though, all this does bring this question to mind. Does having multiple Trees of the same type effective overlap their effect? Like would having to Trees of Time with a overlapping AoE cause plants in the AoE to grow even faster, or would it only read the affect being applied once? Or Does it make the affect just happen faster, like what I think would happen with the Tree of Transformation and the Miner's Tree?
Do, I do want to thank you for trying to help answer my questions. ^w^