In this paper we present an extended model related to reconciliation concepts. It is based on gene duplications, gene losses and speciation events. We define an evolutionary scenario (called a DLS-tree) which informally can represent an evolution of genes in species. We are interested in all scenarios – not only parsimonious ones. We propose a system of rules for transforming the scenarios. We prove that the system is confluent, sound and strongly normalizing. We show that a scenario in normal form (i.e. non-reducible) is unique and minimal in the sense of the cost computed as the total number of gene duplications and losses. Moreover, we present a classification of the scenarios and analyze their hierarchy. Finally, we prove that the tree in normal form could be easily transformed into the reconciled tree [12] in duplication-loss model. This solves some open problems stated in [13].