At 13 months, youre right on the cusp between seat and trailer. Generally, smaller children are better on seats such as the CoPilot Limo ($120), which fits on any good luggage rack (Jandd Expedition; $85). They weigh less, of course, so stability is not affected that much. Rear rack seats are easy to manage, as safe as other alternatives, and cost-effective.
But generally I like trailers. They are a bit more predictable, ride-wise, and the trailer itself provides a bit of extra safety for its occupants. And while I have never ridden in a trailer or a bike seat, I kind a think the trailer has to be more comfortable.
You do pay more. Burleys Honey Bee ($375) seats one or two youngsters, and attaches securely to the bike. With two wheels the Honey Bee is very stable, but of course it is wider than the bike, so you have to take a little care not to hit a curb with the bike trailer in narrow roadways. A trailer such as the Honey Bee also offers weather protection to its occupant, and has places for the inevitable stuff that kids need.
Of course, no matter what you choose, be sure a helmet is part of the equation. Bells Fraction ($40) is a good choice. Be sure you have one, too!