I've taken two different "courses" with regard to bikes with my two oldest kids.
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With my oldest, we bought him a nice (Trek) 16" bike with training wheels when he was 3. Â We figured that, although it was a little big for him, he'd be able to use it longer than a 12" (and it wasn't at all too big for him to ride), and although it was expensive, we had two kids and knew we would have more (we're expecting our fourth and last in September), so we wanted it to be of high enough quality that it would last through several years/riders. Â The bike has held up well, but because it *was* so large, he wasn't able to ride it without training wheels until last summer, when he was 5.5.
With my daughter, we decided to take an entirely different approach: We bought her a cheap, pink, Hello Kitty bike (complete with streamers, doll carrier, and little front-mount-pouch. Â She loved it. Â We got it for her for Christmas this year, just after she turned 4, and she's already riding it without the training wheels, more than a year earlier than my son did (despite not being a particularly coordinated kid). Â Because the bike is smaller, she's able to handle it better, and could manage riding it much sooner. Â
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So, my recommendation is to get a cheap 12"--she's not going to ride it long, so try not to get *too* worked up over the over-the-top styling--for her to learn on, and then a better 16" after she's learned, maybe next summer.
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Oh, and if you do decide to go with a pricier bike eventually, I recommend Specialized a hundred times over Trek. Â My son's Trek is heavier than my or my husband's adult-sized bikes; it's crazy. The Specialized bikes are some of the only aluminum kids' frames I'm aware of, and they have some other nice features as well--my next kids' bike purchase will definitely be a Specialized.