Different covering numbers of compact tree ideals

Abstract We investigate the covering numbers of some ideals on $${^{\omega }}{2}{}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mi>ω</mml:mi> </mml:mmultiscripts> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mrow> </mml:math> associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, $${\mathfrak {C}_{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , remain small in the Sacks model.