Choosing Monte Carlo or a binomial tree

When is Monte Carlo the better pricing method, and when is a binomial (or other lattice) method preferable?

Trees for low-dimensional problems with early exercise; Monte Carlo for high-dimensional or strongly path-dependent payoffs. A lattice handles American exercise naturally and gives fast, smooth prices and Greeks, but its cost explodes with the number of factors. Monte Carlo's error shrinks like $1/\sqrt N$ independently of dimension and copes easily with path dependence (Asians, baskets, LMM), but early exercise needs extra machinery such as Longstaff-Schwartz and Greeks are noisier.