Attack trees are widely used for security modeling and risk analysis. Classically, an attack tree combines possible actions of the attacker into attacks. In most existing approaches, an attack tree represents generic ways of attacking a system, but without taking any specific system or its configuration into account. This means that such a generic attack tree may contain attacks that are not applicable to the analyzed system, and also that a given system could enable some attacks that the attack tree did not capture. To overcome this problem, we extend the attack tree setting with a model of the analyzed system, allowing us to introduce precise path semantics of an attack tree and to define missing attacks. We investigate the missing attack existence problem and show how to solve it by calls to the NP oracle that answers the trace attack tree membership problem; the latter problem has been implemented and is available as an open source prototype.