We use a local Monte Carlo algorithm to simulate lattice trees in two dimensions for the site and bond problem. We investigate the properties of radius of gyration, perimeter-to-site ratio, and vertex degree in a tree, adding some new results in the site problem, compare our results on their noncritical properties with those obtained from earlier reversible and slightly nonreversible algorithms, and combine our determinations with new exact series expansion data. On the controversy surrounding the possible lack of universality for the first confluent singularity for the gyration radius, we feel that conclusions must be guarded.