## 1 Introduction

[2] The branching structure of river networks has been actively studied since the 1960s, to address a broad range of hydrological, geomorphological, and environmental problems. Specifically, the attention of the hydrologic community has been traditionally drawn by the connections between the river network topology and the hydrologic response. An extensive literature therefore exists in this area, starting with the work of *Kirkby* [1976], and *Rodriguez-Iturbe and Valdes* [1979] and followed by numerous other studies reviewed in *Rodriguez-Iturbe and Rinaldo* [1997]. The river network structures have also been shown to constitute a dominant control for other observed natural processes, such as stream biodiversity, riparian vegetation functioning, bed load sediment size distribution, and food web structures [e.g., *Muneepeerakul et al*., 2008; *Power and Dietrich*, 2002; *Kiffney et al*., 2006; *Sklar et al*., 2006; *Stewart-Koster et al*., 2007].

[3] The ability to characterize a river network via its topological properties is a powerful tool to approach the above problems. It allows one to quantitatively describe the connections between the network properties and various observed processes that operate on the network, as well as to perform comprehensive numerical simulations aimed at hypothesis testing and improving the understanding of the network dynamics. For example, the observed scaling relationships between hydrologic (e.g., annual peak flow) and geomorphic (e.g., drainage area, width function peak) variables have been frequently studied through ensemble simulations of synthetic river networks [e.g., *Menadbe et al*., 2001; *Veitzer and Gupta*, 2001].

[4] A commonly accredited property of river networks, based on empirical observations, is the so-called Tokunaga self-similarity (TSS) [*Tokunaga*, 1966, 1978], which constitutes a standard assumption in river network modeling. The Tokunaga self-similar model has two assumptions: (1) the mean number *T*_{ij} of branches of order *i* that merge with a randomly selected branch of order *j* does not depend on the branch orders, only on the difference *j-i*; and therefore *T*_{i(i + k)} = *T*_{k} for any *i*, *k* ≥ 1; and (2) the numbers *T*_{k} obey the exponential relationship *T*_{k} = *ac*^{k − 1} for two positive topological constants *a* and *c*. Since *Peckham* [1995a, 1995b], the Tokunaga model has been gaining increasing popularity, not limited to the hydrologic literature, and constitutes now a ”benchmark criterion” for current network modeling approaches [e.g., *Tarboton*, 1996; *Newman et al*., 1997; *Dodds and Rothman*, 1999; *Cui et al*., 1999; *McConnell and Gupta*, 2008].

[5] The Tokunaga self-similarity assumption of river networks is generally accepted in the literature. However, to the best of our knowledge, a data-based support for TSS has been provided only in a few studies [i.e., *Tokunaga*, 1966; *Peckham*, 1995a, 1995b; *Peckham and Gupta*, 1999; *Tarboton*, 1996] and for a very limited number of basins; most of the subsequent works on the Tokunaga model refer to these studies. Moreover, when the TSS property was reported, data limitations often precluded a rigorous analysis and hypothesis testing. For example, *Peckham* [1995a] noted that the values of *T*_{k} from two real networks “appear to fluctuate about fairly stable values”. While this observation suggested that the considered networks are self-similar, no formal test was applied to investigate whether or not the reported fluctuations were statistically significant.

[6] Studies that statistically confirm (or reject) the TSS property over a range of different climatic and topographic regions are still lacking. An important related question is whether the Tokunaga parameters *a* and *c* show a characteristic value or a range of values. For example, *Cui et al*. [1999] argued that these parameters can be interpreted as representing the effects of regional controls. Moreover, as noticed by *McConnell and Gupta* [2008], it is reasonable to expect some physical restrictions on the values of *a* and *c*, based on the typically observed Horton ratios (i.e., common descriptors of the topological structure of river networks based on a hierarchal ordering of their tributaries) [*Horton*, 1945]; however such restrictions are yet to be explored.

[7] Motivated by this growing interest in the Tokunaga model and by the lack of a rigorous, data-based testing procedure for the Tokunaga self-similarity, the current study has the following main goals: (1) To identify a set of formal statistical methods, that allows to analyze the topology of the river networks and to estimate the accuracy of these methods; (2) to evaluate, on the basis of an extensive dataset, whether the Tokunaga self-similarity assumption for river networks holds across different climatic and topographic regions; (3) to evaluate the range of the Tokunaga parameters (*a*,*c*) in the analyzed Tokunaga self-similar river networks; (4) to investigate whether the distribution of side-branches is geometric as suggested by previous theoretical [*Burd et al*., 2000] and empirical [*Troutman*, 2005; *Mantilla et al*., 2010] studies; and (5) to explore whether the Tokunaga parameters can serve as discriminatory metrics to understand the controls on landscape dissection and river network topology. It should be clear that the focus of this work is solely on the topology of river networks, as we do not consider any geometrical characteristics such as channel lengths or drainage areas. As extensively discussed in section 2, when a river network is found to be TSS, its topology can be completely characterized, in an average sense, by the values of the two Tokunaga parameters, which also define analytically the topological Horton ratios. Moreover, if the distribution of side-branches is geometric (or any other one-parameter distribution), the branching structure of the TSS networks admits also a rigorous probabilistic characterization, based solely on the Tokunaga parameters. Therefore, not only will these results improve our understanding of the topological structure of river networks, but also they will give more confidence (on the basis of the extensive dataset analyzed) in the parameterization of current river network models. As illustrated in section 3, the topological properties are evaluated using well known statistical methods, whose performance is studied using numerical simulations in section 4 to address goal 1. In particular, in section 4 we apply the proposed tests to a set of synthetic TSS trees for which we know a priori the true TSS parameters. We then assess the ability of the proposed tests to accurately estimate these properties, as well as define the confidence level associated with these tests. In section 5 the proposed statistical methodologies are applied to 408 real river networks extracted from 50 catchments across the continental United States (goals 2, 3, and 4).

[8] Recently, *Mantilla et al*. [2010] have analyzed 30 river basins to test a particular form of statistical self-similarity used in the Random Self-similar Network (RSN) model of *Veitzer and Gupta* [2000], as well as to evaluate the range of variability of the RSN parameters. The RSN and Tokunaga models use different mechanisms of generating a self-similar topology: the Tokunaga model uses an appropriate random sampling of side branches [e.g., *Cui et al*., 1999], while the RSN model uses an iterative replacement of randomly sampled network generators [*Veitzer and Gupta*, 2000]. Our results hence complement those of *Mantilla et al*. [2010] in building a solid empirical basis for the theoretical results on river network topology.

[9] The rather wide range of the TSS parameter *c* found from a large number of catchments across the US prompts the question as to what physical parameters might affect the topological structure of a river network and whether the Tokunaga parameters can serve as metrics to explore this question (goal 5). Although a complete answer to this question would require extensive study of climatic, geologic, ecologic, and soil properties of the catchments, we present in this study (section 6) a preliminary analysis of the dependence of the parameter *c* on a range of hydroclimatic variables and report significant dependence. We interpret this result as encouraging, prompting further study on the connection between landscape forming processes and fluvial network topology. Discussion of our results and suggestions for future research are given in section 7.