Table 2.

Analytical and simulation results for critical exponents ρ, ν_path and ν of isolated self-avoiding lattice trees in a good solvent for different spatial dimensionalities d. $\frac{\delta \nu }{\delta {\nu }_{\text{ideal}}}$ is defined as in Table 1. Currently available exact values or best estimates (in boldface) are reported as symbols in Fig. 3, where predictions of Flory theory are shown as lines.

Annealed branching polymers in dilute solutions in a good solvent
d	ρ	νpath	ν	δν	$\frac{\delta \nu }{\delta {\nu }_{\text{ideal}}}$	Technique	Reference
2	4/5=0.8	7/8=0.875	7/10 = 0.7	0.0592	15%	Flory theory	Gutin et al. (1993)38
2	–	–	0.637	−0.0038	-1%	Renormalization group	Family (1980)41
2	–	–	0.6408 ±0.0003	–	–	Renormalization group	Derrida & de Seze (1982)43
2	–	–	0.640 ±0.004	−0.0008	-0.2%	Exact enumeration	Margolina et al. (1984)52
2	–	–	0.6394 ±0.0067	−0.0014	-0.4%	Exact enumeration	Privman (1984)51
2	–	–	0.640 ±0.004	−0.0008	-0.2%	Scanning method	Meirovitch (1987)53
2	–	–	0.644 ±0.004	0.0032	0.8%	Exact enumeration	Ishinabe (1989)54
2	0.74 ±0.01	–	0.637 ±0.009	−0.0038	-1%	Monte Carlo	J. van Rensburg & Madras (1992)55
2	–	–	0.6412±0.0005	0.0004	0.1%	Monte Carlo	Hsu et al. (2005)60
3	9/13=0.692	7/9=0.778	7/13 = 0.538	0.0385	15%	Flory theory	Gutin et al. (1993)38
3	–	–	1/2	–	–	Exact calculation	Parisi & Sourlas (1981)14
3	0.654 ±0.003	–	0.496 ±0.004	−0.004	-2%	Monte Carlo	J. van Rensburg & Madras (1992)55
3	–	–	0.49 ± 0.01	−0.01	-4%	Monte Carlo	Cui & Chen (1996)58
3	–	–	0.500 ±0.002	0.000	0%	Monte Carlo	Hsu et al. (2005)60
3	0.64±0.02	0.74 ±0.02	0.48 ±0.04	0.02	-8%	Monte Carlo	Rosa & Everaers (2016)62
4	5/8=0.625	7/10=0.7	7/16 = 0.438	0.022	13%	Flory theory	Gutin et al. (1993)38
4	–	–	0.425	0.009	5%	Series expansion	Kurtze & Fisher (1979)16
4	–	–	0.450 ±0.035	0.034	21%	Exact enumeration	de Alcantara et al. (1980)42
4	–	–	0.42	0.04	2%	Renormalization group	Parisi & Sourlas (1981)14
4	–	–	0.45 ±0.05	0.034	21%	Exact enumeration	Gaunt et al. (1982)50
4	–	–	0.417	0.001	0.6%	Dimensional reduction	Dhar (1983)45
4	–	–	0.425 ±0.015	0.009	5%	Series expansion	Adler et al. (1988)48
4	0.611 ±0.002	–	0.42±0.01	0.004	2%	Monte Carlo	J. van Rensburg & Madras (1992)55
4	–	–	0.416 ±0.003	–	–	Monte Carlo	Hsu et al. (2005)60
5	11/19=0.579	7/11=0.636	7/19 = 0.368	0.009	8%	Flory theory	Gutin et al. (1993)38
5	–	–	0.359 ±0.004	–	–	Monte Carlo	Hsu et al. (2005)60
6	6/11=0.545	7/12=0.583	7/22 = 0.318	0.003	5%	Flory theory	Gutin et al. (1993)38
6	–	–	0.315 ±0.004	–	–	Monte Carlo	Hsu et al. (2005)60
7	13/25=0.52	7/13=0.538	7/25 = 0.28	−0.002	-6%	Flory theory	Gutin et al. (1993)38
7	–	–	0.282 ± 0.005	–	–	Monte Carlo	Hsu et al. (2005)60
8	1/2=0.5	1/2=0.5	1/4 = 0.25	−0.015	−100%	Flory theory	Gutin et al. (1993)38
8	0.52 ±0.05	–	0.265 ±0.006	–	–	Monte Carlo	J. van Rensburg & Madras (1992)55
Quenched branched polymers in dilute solutions in a good solvent
d	ρ	νpath	ν	δν	$\frac{\delta \nu }{\delta {\nu }_{\text{ideal}}}$	Technique	Reference
3	1/2	1	1/2 = 0.5	0.05	25%	Flory theory	Isaacson & Lubensky (1980)11
3	1/2	–	0.45 ±0.01	–	–	Monte Carlo	Cui & Chen (1996)58
3	1/2	0.87 ±0.03	0.46 ±0.07	0.01	5%	Molecular Dynamics	Rosa & Everaers (2016)62