What is this purple tree?

That's the sleeping tree!

If you find the cave nearby (with one giant in it) you'll find a body with a note that will lead you to Ysolda in WhiteRun. Ysolda, after being shown the note and when asked, tells a tale about the origins of the tree.

Here's the map position of Sleeping Tree Camp, relative to WhiteRun.

Looks like the tree at Sleeping Tree Camp, a giant's camp a fair way to the west of Whiterun, south of Swindler's Den . If you get in the right position near the tree's base, the option "Activate Sap Spigot" will appear. This will give you the potion "Sleeping Tree Sap", which adds 100 points to your health for 45 seconds. Also in the camp is an entrance to Sleeping Tree Cave, which holds a clue about something you can do with the sap...

The tree is in fact a distant cousin of the hist trees from black marsh,ysolde comments on a tale that the tree is from some "floating island" also known as Umbriel, the tree also matches the description as presented in the novels by Greg Keyes, and like the hist, is capable of producing hallucinogenic sap that alters the users mind.