Hello,
The tree in your picture is not a Chinese elm. From what I can see it is a Lingstrum Privet. You can easely identify your trees using :
www.bonsaiempire.com/tree-species/bonsai-identification
or resources from other sites.
First thing I would look into is if it has everithing it needs: Water, ventilated area (fresh air), temperature, nutrients (did you fertilise it).
Also check if it is root bound in the pot, if it is, you may need to change your watering habit. I have a tree that is root bound and it needs to be watered more frequently and thuroughly because the soil will not retain that much water. Take care to not water it too much though.
Depending on your area, if it is not too cold, I would advise to move the tree somewhere outside, or in a balcony. I have a ligustrum and mine is doing great outside, when I first bought it I was keeping it inside and I had some problems with it. The fresh air and temperature did wonders.
Fire or heat, in general, tend to dry the air very badly, that may be another cause why your tree is losing some of it's leaves. If you can not move the tree outside, try to mentain the level of humidity and the temperature (avoid cold-hot-cold mix).
From what I experienced, Ligustrums are pretty hardy and will wistand almost anything as long as it has an appropriate environment and is watered properly.
Note: I am a beginner myself and I tried to tell you what I would do and what I think could help. As far as I know it is not an exact science, your problem could be from lack of watering or over watering, there are no quick fixes, you just need to provide the proper environment and the tree will mostly take care of itself. Also I found that trying different things helped me learn a lot.
