On the Shape of Trees: Tools to Describe Ramified Patterns

The branching structure of diffusion-limited aggregates

“…For DLA in two dimensions, for which y(η) is known, solving eq. ( 14) gives α ≈ 0.628, or B ≈ 4.92, in reasonable agreement with the numerical result of B ≈ 5.2 [5], [6]. In higher dimensionalities, we must rely upon model Z.…”

“…Physicists have concentrated on understanding and predicting the fractal and multifractal properties of DLA [3], [4]. However, several authors have advanced an alternative approach, focussing on the topological self-similarity of DLA clusters, as measured by quantities typically used to describe river networks and other branched structures [5], [6]. This letter is devoted to the computation of these quantities, using the "branched growth model" which this author and his collaborators have exploited to compute a number of properties of DLA [7].…”

“…River networks are often topologically self-similar, with B i ≈ B and 3 < B < 5, even though they do not display conventional geometrical fractality, since they extend close to every point in a basin. DLA clusters in two dimensions also exhibit topological self-similarity, with B i ≈ 5.2 for i > 1 [5], [6], [9].…”