What are the best types of wood for building planter boxes? How thoughtful should they be for vegetables?

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The depth depends on the vegetable. Large plants like tomato, cucumber, squash etc requirement at least 12 inches.
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Answers:    Cedar or redwood will hold up the best.
As far as depth, it depends on the size of the vegetable. For things like radishes, spinach, green onions, bush beans, and so on, 6" is probably so-so. For leeks, onions, peppers, carrot, cukes etc. you'd want at least 10-12 inches. Tomatoes, squash, you'd involve at least 12-16 inches.