Assuming there aren’t any restrictions, and you decide that a tree has to go, it is far better to deal with it while the garden is being gutted than leave it for a while until it gets out of hand. Tree removal makes a lot of mess and temporary damage to the surrounding area, and lopping back the branches to a more suitable length is no real solution. Lopped trees look ugly and the natural shape, which is often the most attractive thing about them, is permanently spoilt. Also, depending on the rate of regrowth, you will probably have to remove the new branches annually to keep the tree in check — lopped trees always grow faster than ones left alone, until the top growth becomes proportionate to the root system again.
Check list for tree removal
Very large trees are best dealt with by reputable tree-felling contractors. A badly felled tree can do irreparable damage. Felling contractors have the machinery for the job, the expertise and protective clothing and harnessing. Personal recommendation is the best way to find a firm specializing in this sort of work. There are probably more unskilled, untrained cowboys in this trade than any other. By the time you realize you have employed one of them, it may be too late. Ask to see their certificate of public liability insurance before entering into any contract.
Some felling contractors will also remove the stump for you if required. If the garden is large enough, they may drag it out with a tractor and chain, or lift it out with a JCB. Otherwise, there are stump-boring machines which will drill most of the stump out. Try and ensure that as much of the old root as possible is removed as certain diseases (e.g., bootlace fungus) can occasionally start this way in a garden.
If the stump is not removed it will sprout again and so should be killed off. One method of doing this is by drilling deep holes in it and filling them up with a brushwood killer, then covering the stump with waterproof material to prevent rain diluting the chemical. The stump will eventually die and rot.
Another alternative is to leave several feet of stump sticking out of the ground, kill it off as before, and grow climbing plants up it. This has a lot of drawbacks, from looking rather odd if in the wrong place, to harbouring pests and diseases, and eventually blowing over when it is rotten and top heavy, so should only be used as a last resort.
Smaller trees can be felled in stages using a hand-saw or small chain saw, which should be quite easy for a handyman. The stump can usually be dug out by hand or covered up as described above. Small trees and stumps can sometimes be winched out using a caravanner’s winch.
Small branches and twigs can be burned if convenient or shredded, larger branches and trunks sawn up and used as firewood or burned. If burning would cause problems and the timber is not required for firewood, a skip is the best way to get the rubbish off the site. Large roots do not burn well except when they have been exposed to the air and dried off for a considerable time, so are best put in a skip.