Is it true that ...

The closest tree is a random tree?

No. Not with any sampling scheme that we are aware of. (We are assuming that you are at some random point in the forest and are selecting some tree around you).

If you are using Fixed plots or strips, you are likely to be nearer a large tree (because they have more space around them, and you are more likely to fall into it).

If you are using Variable Plot Sampling then you have a weighting that is proportional to basal area initially. Even if you wanted that, you still have a problem. With an angle gauge you are more likely to be near a small tree.

This is because once you have chosen the trees (which should be equally likely), you are likely to be closer to a small tree because the circle around it is smaller – even though the sum of those small circles may be the same as the single large circle. That one is a tricky bit of logic, and not well known.

If you have to pick a random tree from a group, go to the trouble of picking a random number (many calculators can do that for you). If you have 7 trees, pick a random number between 0-1 with the calculator, multiply by 7, add 1, and use the number in front of the decimal.

The idea of "random" doesn’t mean that you did not mean to do something wrong. It is a very specific statement of probabilities. Don’t say that something "is random" when you mean that you did not care what the outcome was and were trying to be fair. It’s nice to know – but it does not make things "random". You should only say this when you are sure that the outcome was appropriate.