The next specific case is forest. Forest grows on one of four main type of terrain: hills, grasslands, plains and tundra. It's generating some of additional output functioning as a natural improvement (gives +1 to Production) what makes it one of the most useful early-game terrains. In addition, every city which has it in reach takes +0,4 health bonus for one forest.
The most important feature of the forest is the possibility of removing it. There's a worker action called "chop trees" which becomes available after inventing bronze working. Chopping the trees and removing the forest gives our nearest city a sudden Production bonus of at least 20-30, what can be extremely useful in situations when you are looking for the opportunity of rushing building of some units or buildings. The classical use of this trick is rushing a settler or Wonder, but it's useful in variety of situations.
So called chop rush, especially in extreme version is very a controversial strategy because you gain very significant addition to Production output for the price of loosing really useful terrain.
It's impossible to plant trees, and even the fact the forest is constantly expanding doesn't change the situation, as its growth is very slow and you haven't got any influence when and where it happens.
There are some ways of limiting the negative influence of this strategy for your empire. First of all, it is possible to chop trees outside of city reach and still receiving a Production bonus. It even works to cut trees growing outside your borders (but not in another empire's territory) and still receive this bonus. So, when you have a huge forests somewhere you don't plan to expand you may chop rush without any negative consequences.
The second situation when we can cut the trees without any remorse is building any improvements (except Lumbermill) at forest title. You simply give your workers an order to build improvements and he will cut the trees and build the structure in this same time. You will have a beneficial influence from this tile and bonus Productions - all in one.
The jungle is in some way opposite to forest because of its subtract amount of output from specific tile and gives health penalty. It doesn't give any benefits from being cut down except, of course, being removed.