The idiom, “you can’t teach an old dog, new tricks” relates to the idea that a mature tree cannot be trained for structure without sacrificing its health and vigor. That being said, two types of mature trees exist; trees that have been maintained and trees that have been unkempt. Maintained mature trees are simple to care for as they already have proper structure and form. You can expect that the needs of properly pruned trees will be the occasional removal of dead branches, raising the crown, and encouraging healthy branch angles. In comparison, the unmaintained tree will take more time and effort to correct.
Shade trees lacking competition found in a forest setting begin to not only grow in height, but in width. Trees of this category still need basic crown thinning and raising, and also need to be structurally pruned slowly, by removing no more than 25% of the canopy with each pruning session. If you have inherited an unmaintained tree, just have patience, and avoid topping.