Irredundance, secure domination and maximum degree in trees

It is shown that the lower irredundance number and secure domination number of an n vertex tree T   with maximum degree Δ⩾3Δ⩾3, are bounded below by 2(n+1)/(2Δ+3)2(n+1)/(2Δ+3)(T≠K1,Δ)(T≠K1,Δ) and (Δn+Δ-1)/(3Δ-1)(Δn+Δ-1)/(3Δ-1), respectively. The bounds are sharp and extremal trees are exhibited.