Construction and properties of fractal trees with tunable dimension: The interplay of geometry and physics

In this paper, we emphasize three different techniques for the growth of fractal trees with a desired fractal dimension Df. The three different growths are due to the influence of (i) stretched branches, (ii) dead ends, or (iii) a variable branching rate. Several examples are given. We point out that geometrical and physical properties (skeleton dimension, percolation exponents, self-avoiding walk) of fractal trees depend strongly on their type. The most striking result is that the critical exponents at the percolation transition are nonuniversal since they depend on the tree type. The critical exponents depend on Df for trees of types (ii) and (iii).