Please explain which is the best URL structure and why:
This is a property site. Is it best for the bots to break the properties/URL's down in categories?
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Regarding SEO or bots, there is no difference between these two URLs. To implement good URL format, you need to think about your users. You also need to be logical, i.e. adding a subdirectory means you define a new set of webpages.
For example, adding
/manhattan/ in your URL means all webpages under this subdirectory will present something about Manhattan, apartments for example.
The best practice for websites architecture like yours is to create categories by functionality and cities. Therefore, users could easily search apartments they are looking for. You thus can use URL in this format (proper way):
example.com/apartments-to-let/manhattan/ example.com/apartments-to-let/boston/ example.com/apartments-to-sell/los-angeles/ ...
I agree with Zistoloen. It doesn't matter for SEO. As long as you have your keywords in the URL w/out going overboard and those keywords describe the page, you'll be fine - http://moz.com/learn/seo/url
The only thing I would add is to do preliminary research to make sure you're choosing the best keywords for your landing pages. It looks like you are centering an entire section of your site to "Apartments to Let". That seems too competitive and not something that would bring in good leads.