Trees are great. They produce the oxygen we need to survive and keep our air clean; they provide vital habitats for wildlife; they give us healthy, delicious food--and that's not the half of it. One of the oldest living things on earth is a tree in Sweden, thought to be almost ten thousand years of age. Humanity cannot do without trees, and it's important to respect and protect them.
That said, there are some trees that are simply in the wrong place. The Queensland Government describes trees as "a common cause of disputes between neighbors", and there is some evidence that trees close to buildings can cause structural damage and other problems. It may also simply be that you are remodeling your garden, and there's currently a tree where you'd like a lawn to be. If you have a problematic tree that you need to have removed, what can you do to ensure that you're behaving in a respectful and environmentally friendly way?
Have the tree removed at the right time of year.
If your tree is used by nesting birds, it's extremely important that you don't have it removed until those birds have fledged and the nest is no longer in use. As well as simply being the right thing to do, in some parts of Australia, it's illegal to disturb a nest.
Err on the side of replanting as often as you can.
It's not always practical to replant a tree you're having removed; you might not have space for it, it might be too large to move, or it may simply not survive the operation. If it's a possibility, though, it's always worth looking into. If your smallish tree is simply in the wrong place, there might be a perfect home for it in another part of your garden.
Recycle and re-use as much as possible, and dispose of what's left responsibly.
Most tree surgeons will take large trees away with them, but smaller trees don't necessarily need to be disposed of in such a way. Leaves and small twigs are great for your compost heap, and you may be able to find practical uses for much of the wood. If it's safe and legal in your area to do so, you can burn what's left cleanly on a bonfire in your own garden--but check the law first. Many parts of Australia have localised laws about bonfires, or declare Total Fire Bans on days when risk of spread is too high.