Independent double roman domination number of a tree in terms of its 2-independence number

Let [Formula: see text] be a simple graph. An independent double Roman dominating function (IDRDF) on a graph [Formula: see text] is a function [Formula: see text] having the property that first if [Formula: see text], then vertex [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; second if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor [Formula: see text] with [Formula: see text] and third the set of vertices assigned with positive numbers is independent. The weight of an IDRDF [Formula: see text] is equal to [Formula: see text]. The independent double Roman domination number [Formula: see text] is the minimum weight among all IDRDFs of [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. In this paper, it is shown that both parameters are incomparable, in general. Also, we prove that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.