On the topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index $r=\max{(i,j)}+\delta_{i,j}$ has been shown to be relevant to a number of physical (such at diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees) and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of $n$ leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of non-linear functional equations, we are able to give a double-exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos correcte

Topological self-similarity on the random binary-tree model

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented

Self-similar community structure in organisations

The formal chart of an organisation is designed to handle routine and easily anticipated problems, but unexpected situations arise which require the formation of new ties so that the corresponding extra tasks can be properly accomplished. The characterisation of the structure of such informal networks behind the formal chart is a key element for successful management. We analyse the complex e-mail network of a real organisation with about 1,700 employees and determine its community structure. Our results reveal the emergence of self-similar properties that suggest that some universal mechanism could be the underlying driving force in the formation and evolution of informal networks in organisations, as happens in other self-organised complex systems

Effects of watershed land use on nitrogen concentrations and δ15 Nitrogen in groundwater

Author Posting. © The Authors, 2005. This is the author's version of the work. It is posted here by permission of Springer for personal use, not for redistribution. The definitive version was published in Biogeochemistry 77 (2006): 199-215, doi:10.1007/s10533-005-1036-2.Eutrophication is a major agent of change affecting freshwater, estuarine, and marine systems. It is largely driven by transportation of nitrogen from natural and anthropogenic sources. Research is needed to quantify this nitrogen delivery and to link the delivery to specific land-derived sources. In this study we measured nitrogen concentrations and δ15N values in seepage water entering three freshwater ponds and six estuaries on Cape Cod, Massachusetts and assessed how they varied with different types of land use. Nitrate concentrations and δ15N values in groundwater reflected land use in developed and pristine watersheds. In particular, watersheds with larger populations delivered larger nitrate loads with higher δ15N values to receiving waters. The enriched δ15N values confirmed nitrogen loading model results identifying wastewater contributions from septic tanks as the major N source. Furthermore, it was apparent that N coastal sources had a relatively larger impact on the N loads and isotopic signatures than did inland N sources further upstream in the watersheds. This finding suggests that management priorities could focus on coastal sources as a first course of action. This would require management constraints on a much smaller population.This work was supported by funds from the Woods Hole Oceanographic Institution Sea Grant Program, from the Cooperative Institute for Coastal and Estuarine Environmental Technology, from Massachusetts Department of Environmental Protection to Applied Science Associates, Narragansett, RI, as well as from Palmer/McLeod and NOAA National Estuarine Research Reserve Fellowships to Kevin Kroeger. This work is the result of research sponsored by NOAA National Sea Grant College Program Office, Department of Commerce, under Grant No. NA86RG0075, Woods Hole Oceanographic Institution Sea Grant Project No. R/M-40