I think what you are seeing on the plant is sooty mold, a fungus that develops on "honeydew" that is secreted from insects such as aphids, scales, and mealybugs. These are all common pests of indoor plants. They do their damage by sucking juices from the plant and excrete a sticky material called honeydew. Aphids are fairly easy to see. They are soft-bodied insects. Scales can be visible as well, but can be a bit misleading because they can simply look like small brown bumps. Mealybug is quite obvious, looking like bits of cotton. In any case, they need to be controlled. Depending on the size of tree, taking it to a tub and showering the plant is a safe way to clean up the leaves. Insecticidal soaps are available at garden centers. Follow the directions. Other pesticides are available, but treatment would need to be done out-of-doors, and this is impossible at this time of year. Usually multiple treatments are needed since there the eggs are laid over several weeks. You need to control the insects in order to control the black mold.