Eucalyptus trees are not native to the United States. They are immigrants from Australia. Eucalyptus grow quickly in warm, moderately dry climates such as found in much of California and the Southwest. If you decide to remove one of these trees from your property start by talking to your local city planning office to find out if you need a permit to cut a tree. If so, there may be fees involved.
Estimate the height and width of the tree and decide if the entire tree can be removed at once without the tree falling on buildings, power lines or damaging any other property. A quick way to estimate a tree's height is to cut a stick as long as your arm and then hold the stick in front of you at arm's length. Back up until the entire tree is hidden behind the stick. Measure from that spot to the base of the tree to determine its height. (See Resource section below for additional information on estimating a tree's height). If the tree is too tall and needs to be topped before it can be cut down it is recommended that you hire a licensed professional to do your tree topping as this can be a very dangerous activity.
Notch the tree on the side that you want it to fall. Start your notch by making a horizontal cut at about waist height straight into the trunk of the tree with your chainsaw. Cut approximately half way through the tree. Be patient and cut slowly as eucalyptus wood is very hard and it is easy to overheat the chain and/or dull it.
Start your second horizontal cut approximately 12 to 18 inches above your first horizontal cut. This time cut downward at approximately a 45-degree angle until your second cut reaches your first horizontal cut approximately half way through the tree. Remove the wedge of wood that these two cuts created. You now have an open notch in the side of your tree that is aimed in the exact direction you want the tree to fall. When cutting your notch take breaks to allow your chain to cool as it cuts through the hard eucalyptus wood.
Plan your escape routes. You should have more than one direction to run just in case the tree does not fall exactly as planned.
Make your final cut straight through the trunk of the tree from the side opposite the notch. This cut should be approximately 2 to 3 inches above the center of the "V" of the notch. Before you have cut all the way through the tree to the notch the tree will begin to fall. It should fall in the exact direction that your notch is pointing. Run away from the falling tree at a 45-degree angle (not directly behind the falling tree).
Cut the tree into manageable pieces with your chainsaw for proper disposal.