Tree removal
There are Council laws that prohibit the removal of trees. It is best that you speak to the Council before removing the tree. If possible, I would do the following:
1 - Keep a record of any photographs or reports showing that the trees were rotting;
2 - As a matter of good faith between yourself and the Council you may choose to inform them of the removal of the trees (however this may lead to a prosecution against you);
3 - You may plant further trees as a replacement. If the Council find out that the trees were removed illegally, it would be helpful if you had taken steps to replace them.
Overall, many Council's will not prosecute a person who has removed rotten trees that are not indigenous particularly in circumstances where the owner was ignorant of the law. Nevertheless, you risk prosecution.
If you choose to replace the trees you could contact the Council for a list of trees that are indigenous to your area. You may also give consideration to fining out whether the tree/s that you select are prone to dropping limbs. Whether you choose to inform the Council or not, I would plant replacement trees. This will bode well for you if the Council choose to pursue the matter.