## 1 Introduction
Drainages obtain long range interaction, multi-fractality and present a memory of the geological and physical processes involved in their evolution. Since memory and fractality are two of the key components of any geophysical process [36; 37; 38] we can use a current generalization of Boltzmann-Gibbs (BG) statistical physics, referred as non-extensive statistical physics (NESP) [27; 28; 36] to justify the hierarchy pattern presented in drainage area distribution. The advantage of considering the Tsallis distribution [28] is that, based on the fundamental principle of Tsallis entropy, scaling laws observed in phenomena that present fractality, long range interaction and memory effects [27; 28] could be interpreted. We note that recent applications to solid earth physics (in regional or planetary scale) [31; 32; 33; 37] and to natural hazards [29; 30; 38] supports the applicability of non-extensive statistical physics in complex geosystems. To our knowledge, this is the first time that Tsallis entropy is used to express and interpret the drainage basins area distribution, within an effort to present scaling laws as extracted from first principles and not in an empirical basis. Scaling theories play an important role in quantification of scale invariance in geosystems, since the appearance of power-law behavior has been seen as the signature of scale invariance. Our motivation is to suggest a new view of scaling laws observed in drainage basins of Greece in terms of Tsallis entropy. A connection between the observed power law exponent and the Tsallis entropic parameter is suggested. To demonstrate the hierarchy observed in drainage area statistical pattern, data sets for all drainages all over Greece, along with subsets of drainages in the internal and external Hellinides, that are the main tectonic structures in Greece, are constructed. Since the topographic relief is a crucial parameter in drainages formation, we classify them according to the mean value of the Topographic Position Index (TPI) and the mean elevation. Finally, the observed hierarchy within each one of the sub-datasets, is interpreted according to non extensive statistical physics. We clarify that the zonation used is the external input in our analysis, as introduced by geotectonic and geomorphological critiria. We note that the selection of TPI and of mean elevation is based on their simplicity as geomorphological measures that control a number of phenomena (e.g., erosion). The remaining of this presentation is organized as follows. A procedure of data extraction using Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) Global Digital Elevation Model Version 2 (GDEM V2), images is presented in Section 2. A brief presentation of the NESP formalism will be given in Section 3, followed by a presentation of the hierarchical drainage basin analysis applied power law formulation and along with NESP expressions. The analysis as presented in Section 4, will focus on the drainage data set of Greece along with the subsets created by the classification of drainages in the geotectonic frame of internal or external Hellenides or using classifications based on topographic criteria as that of mean elevation or the mean Topographic Position Index (TPI). Section 5 is devoted to the discussion of the results and to the presentation of a possible origin of the deviation from power law observed in a number of cases for large drainage area. Finally, it shall be demonstrated that drainage systems are sub-additive systems with significant long range interaction where non-extensive statistical physics could be used to understand the observed hierarchical processes.