Zero-based array implementation with logarithmic insertion time Normal zero-based arrays (ie not those with a sort order) have constant lookup time, but linear insertion time.

For a specific problem I was musing about a balanced tree that would allow for an zero-based array implementation with logarithmic time for all of the following operations: insertion, removal and finding the next and previous element.

I'm reasonably sure it should be straight-forward to implement: The non-leaf nodes would have to store the count of elements in their left branch respectively.

Such a data structure should be useful in text editors, as the user obviously can insert text anywhere in the document and yet the editor is supposed to render a screenfull of text around a given location quickly as well.

My question: What's this data structure called? Is there a buzzword I can google? (Such as "B-tree", "red-black tree", etc. for key-ordered search trees?)
All tree data structures that use pointers can be re-worked to use array indices as well, and that is probably your best bet. If you come across a design that uses NULL pointers, use the array index $-1$ to represent NULL.

In general, the best kind of structure to look for (and to arrayify if that is what you want to do) is a rope, which is a stronger kind of string (see Wikipedia). Ropes are designed for exactly this purpose.