Tree surgeons use a specific vocabulary for pruning operations, and the differences are not cosmetic. Knowing which one you are being quoted for is the best single check on whether a contractor knows their trade.
Crown reduction
Reducing the overall size of the crown, in height, spread or both, while retaining the tree's natural form.
Each branch is shortened back to a suitable lateral growth point, a smaller branch already growing in the right direction, which then takes over as the new leader. The tree looks like a smaller version of itself rather than a mutilated one.
MyJobQuote puts crown reduction at £400 to £900.
A reduction is specified as a measurement, for example a two-metre reduction, and a competent arborist will tell you if what you have asked for is more than the tree should take in one operation.
Topping, and why to refuse it
Topping means cutting the main stems back to stubs, with no regard to growth points. It is quick, cheap, and bad for the tree.
The tree responds to losing most of its foliage by throwing out dense, vigorous regrowth from around the cut. Those shoots are weakly attached, growing from the outer layers rather than from a proper branch union, and they are far more likely to fail in wind as they get heavier.
Meanwhile the large stub wounds cannot seal properly and become entry points for decay. So a topped tree ends up bushier, more hazardous and shorter-lived, and needs cutting again within a few years.
If a contractor offers to top your tree, get another quote.
The other operations
Crown thinning removes a proportion of the smaller branches throughout the crown to let more light and wind through, without changing the tree's height or shape. Useful for reducing shade and wind resistance.
Crown lifting removes the lowest branches to raise the height of the canopy, typically to clear a path, a driveway or a view. Straightforward and among the cheaper operations.
Deadwooding removes dead, diseased or broken branches. Usually the first thing worth doing on a neglected tree and often the only thing needed.
Pollarding is a management system, not a one-off. Cutting a tree back to the same points repeatedly on a cycle, common on London plane street trees. It has to be started when the tree is young and continued, so it is not a treatment to begin on a mature tree.
Getting the specification right
Ask for the work to be written down in these terms, with measurements. "Cut the tree back a bit" is not a specification, and two contractors will price and execute it differently.
If the tree is protected by a TPO, the council will want the work described in exactly this vocabulary on the application, and a vague application gets refused. A contractor who cannot write a proper specification is a contractor who will struggle to get you consent.
How much is too much
Removing more than about a quarter of a mature tree's crown in one season is a lot. Trees fund regrowth from stored energy, and taking too much at once stresses them into the weak, dense regrowth described above.
Where a large reduction is genuinely wanted, staging it over two or three seasons produces a better and safer result.