Lagrange formula for differential operators and self-adjoint restrictions of the maximal operator on a tree

The paper is devoted to linear differential operators defined on a tree. We aim at obtaining complete descriptions of well-posed restrictions of a given maximal differential operator on a tree. In this paper all self-adjoint restrictions of the maximal operator and also all the invertible restrictions of the maximal operator are described. We also present the Lagrange formula for a differential operator on a tree with the Kirchhoff conditions at its interior vertices.