Does Family Tree PHP has a xml sitemap function for Google and other search engines?

Yes, Family Tree PHP does have such a function. You can access the sitemap via /index.php?a=sitemap.xml or (if you are using search engine friendly urls) just via /sitemap.xml.

You can submit the url of your sitemap via Google Webmaster Tools. This will enable Google to find all of your pages easily. Other search engines have similar options.

However, if your site is still very new and has already a big database, resulting in many pages, submitting your xml sitemap at Google can result in Google banning your site from the search results. Google likes sites that grow slowly over time, because that’s how sites normally grow. A site that immediately starts with hundreds of pages will be suspicious in the eyes of Google and treated as a potential spam site. So, beware!