[Note: A few people have turned up here looking for information on banana trees walking. A better source than my ramblings is the wikipedia entry on bananas]
While in the Canary Islands, I discovered that bananas haven't had sex for 10,000 years. Since I've already gone this far down the route of investigating bananas, I might as well continue, so here's the full story.
Wild banana fruit are full of seeds, and generally not very good for eating. On the other hand, this is a result of them reproducing sexually. They also reproduce asexually, by growing a new trunk or stem for each bunch of bananas (which is why banana trees "walk" over time, as the old stem dies and a new one takes over). If you're a neolithic New Guinean proto-farmer in 8,000 BC you can transplant these new stems, to create new banana trees. But since the fruit is full of seeds, why would you?
Well, fairly obviously, occasionally banana trees grow which are sports; infertile plants with no seeds in their fruit. Usually these would be the exception, but neolithic farmers discovered one (or more likely, several at different times; 8,000 BC is the earliest date. The one's you eat probably weren't domesticated until 4-5,000 BC), transplanted stems, and the rest is history.
History with no sex in, which makes it clean, wholesome fun for the entire family.
[Edit: Some auto-content-harvester stole my last post and put it on their website to make it look like they are a real blog with real people rather than a robot-thing that just exists to lead people to products and services. In the near future my posts will try and avoid this by being even less likely than usual. I should probably have checked that wildbananasex.com doesn't exist, but, you know, who has the time?]
 If you find evidence to the contrary elsewhere on the internet, I really don't want to know.
 They're not really trees, but I'm not doing the research for this post.