There are numerous exotic animals that love mud. Elephants are known to deposit mud over their bodies to cool themselves down, and some macaw parrots actually eat mud for its mineral content. And yet you're not going to see these beautiful creatures if your suburban Australian backyard was to turn into a giant mud puddle. While you can't control the weather, you can control how your backyard withstands the weather. This is something to think about when you decide to remove a tree from your backyard. Without a contingency plan, the removal of a tree has the potential to turn your property into a mud puddle after heavy rain. This doesn't mean that you need to leave an unwanted tree in place. It just means that you might need to replace it with something (or leave a relevant part of the tree where it is).
Total removal of the tree leaves you open to soil erosion. This can be particularly damaging if the tree was on a slope or a slightly angled piece of ground. The tree's root system would have kept the soil in place, but once this has been removed, it doesn't take much rain for the newly loose soil to be washed away, perhaps culminating in that backyard mud puddle. Leaving the stump intact is one way to overcome this.
Working with the Stump
Whether the tree is cut down as a single unit or reduced in sections, the stump can be left in place. Leaving the stump in the earth will help to prevent soil erosion, and while the root system might die once the trunk and canopy have been removed, it will still assist to compact the soil. If the simple aesthetics of leaving a tree stump in your backyard doesn't appeal, you could consider covering it with a climbing plant of your choosing.
When the Stump Is Removed
If you want the tree removed in its entirety, then your tree removal company will also be able to take care of stump removal. It can then be replaced with a sapling, ideally one that will quickly put down a root system of a comparable depth to the tree it's replacing. Immediate planting is preferred while the soil is aerated. Be careful not to replace the stump with a tree that has a disproportionately deep root system, as this can cause issues once the tree matures.
Grasses and Rocks
You also have the option of covering the newly exposed site with a tussocky natural grass, which will contain the soil. They are generally fast growing and easy to maintain. Of course, the site doesn't need a new tree or plant. Gravel and ornamental rocks will keep the soil in place, even until such time as you make a decision as to what to plant. Be wary of gravel on a steep slope, as this can easily wash away (along with the soil) during a heavy rain.
In order to avoid a backyard mud puddle, it's really important that you don't remove a tree without having a plan about what you're going to put in its place.