Christmas trees are popular holiday decorations. Occasionally, the carefully selected tree will harbor a number of insects and small arthropods. These may be a nuisance but are typically not harmful. Penn State has a guide to commonly encountered insects and how to address them.
First, it is useful to not introduce insects. A tree should be inspected before bringing it indoors. Some vendors have shakers to physically remove loose needles and twigs. Shakers will also dislodge small insects and birds nests that can harbor insects and mites. Egg cases of praying mantids are occasionally attached to trees and should be removed and left outdoors. Mantids overwinter as eggs and time their emergence to coincide with prey. Brining them indoors can trigger the development and hatch of mantids that will starve for lack of prey.
Most insects cannot survive in the dry barren indoor climate. A simple effective method of removing them is by vacuum cleaner. Spraying a tree with insecticide is not advised as some aerosol propellents are flammable or can increase the flammability of a cut tree.
My most memorable experience with insects and Christmas trees involved a potted live tree. In Spring, we place our live potted tree outdoors and bring it inside when temperatures free in Fall. Shortly after moving the pot from the cold outdoors to the warm indoors, our house was buzzing with dozens of blow flies. Why? The tree had been next to a window. Late in the year, a bird had flown into the window and bounced into the pot where it died. Blow flies colonized the bird and pupated in the soil. Seasonal factors halted their emergence and the pupa were dormant awaiting the arrival of warm Spring weather. The warm indoor temperatures caused development to resume and the blowfly adults to emerge. Fortunately blow flies are slow and eliminated in less than a day.