I wasn't really sure how to model the z-coordinate as a function of height. I had a hunch it would be the case, but I couldn't actually figure out how to do it. Thanks to the smart people in the thread for coming up with the equations for it!

Tim, does the fact that the spiral is also travelling in the Z-direction change the length? Is there a distortion due to the 3-D nature of the spiral? I honestly don't know, which is why I added the helpful hint; in case someone could answer that, did the full 3-D calculation, and came up with a more correct answer. If the length of the 2-D spiral is equal to the 3-D one, then your equation is correct; that's actually how I calculated it originally before realizing that I could just use the base of the tree as a variable instead of both the height and base. Does anyone who can actually do this sort of calculation want to chime in here?