Rods to self-avoiding walks to trees in two dimensions

The mean-square radius of gyration 〈${\mathit{R}}_{\mathit{G}}^{2}$〉 and a shape parameter \ensuremath{\Sigma}=〈${\mathit{R}}_{\mathit{G}\mathrm{min}}^{2}$〉/〈${\mathit{R}}_{\mathit{G}\mathrm{max}}^{2}$〉 are studied as a function of the number of bonds, bends, and branches of self-avoiding lattice trees on the square, triangular, and honeycomb lattices. We identify the universality classes, and exhibit the crossover scaling functions that connect them. We find (despite doubts recently raised) that there is a universal crossover from rods to self-avoiding walks, embodied in 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}^{2}$U(Nw), where w(z) is an appropriately chosen nonlinear scaling field reducing to the stiffness fugacity z as z\ensuremath{\rightarrow}0 ; that ``rigid trees'' (which are bond clusters that branch but do not bend) are in the same universality class as branched polymers or free trees; that the crossover from rods to rigid trees has the universal form 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}^{2}$W(${\mathit{Ny}}^{2}$), where y is the branching fugacity; and that the crossover from self-avoiding walks to branched polymers has the universal form 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}_{\mathit{s}}^{2\ensuremath{\nu}}$Y(${\mathit{Ny}}^{\mathrm{\ensuremath{\varphi}}}$), with ${\ensuremath{\nu}}_{\mathit{s}}$=3/4 and \ensuremath{\varphi}=55/32.