Recently, Mondal et al. introduced several novel topological indices based on neighborhood degree sum of vertices such as neighborhood version of the first, second and third Zagreb indices. The neighborhood version of the first, second and third Zagreb indices is defined as \(\sum _{x\in V(G)}S^2_{G}(x)\) , \(\sum _{xy\in E(G)}S_{G}(x)\,S_{G}(y)\) and \(\sum _{xy\in E(G)}[S_{G}(x)+S_{G}(y)]\) , respectively, where \(S_{G}(x)\) is the neighborhood degree sum of the vertex x which is the sum of degrees of all of its neighboring vertices. In this paper, the minimum values of neighborhood version of the first, second and third Zagreb indices are determined within the set of trees having given order and maximum vertex degree. We further establish the corresponding minimal trees.
