Yes, it is possible for Christmas trees to trigger allergies. This is especially true for juniper and cedar which can bring pollen into your home and trigger allergies. The main problem however is mold.

Freshly cut tree can breed mold spores. The spores grow all over the tree and when the spores are released into the air,  inhaling them into the nose can trigger allergies.

You can use a leaf blower on the tree before dragging it into the house. You may also want to rinse the outside of the tree with water using a garden hose and let it dry thoroughly before setting it up inside.

If someone is allergic to molds , you may want  to use a clean , artificial tree or keep the fresh one for less than a week to minimize the spores.

You also have to remember that artificial trees  and other ornaments can contribute to allergies if they are left out for too long  or stored improperly. Clean your decorations with bleach solution when they get dirty. It is important to make sure they’re clean and dry before you put them away in plastic containers.