Multiplicative Sombor index of trees

For a graph Ω, the multiplicative Sombor index is defined as ∏SO(Ω)=∏ab∈ E(Ω)√d²_Ω(a)+d²_Ω(b), where d_Ω(a) is the degree of vertex a. Liu [Liu, H. (2022). Discrete Mathematics Letters, 9, 80–85] showed that, when T is a tree of order n, ∏SO(T) ∏SO(Pₙ)=5(√8)ⁿ⁻³. We improved this result and show that, if T is a tree of order n with maximum degree D, then $$∏SO(T) \{{array}{ll} (5({{D}}^2+4))^{{{D}}{2}}8^{{n-2{{D}}-1}{2}} & { if}\;{{D}}-1/2,\\[2mm] ({{D}}^2+1)^{{2{{D}}+1-n}{2}}(5({{D}}^2+4))^{{n-{{D}}-1}{2}} & { if}\;{{D}}>n-1/2. {array}. $$ Also, we show that equality holds if and only if $T$ is a spider whose all legs have length less than three or all legs have length more than one.