Where does East meet West?

[Don]

Connection made!
After a brief afternoon outing measuring exemplary spruces and pines, the foursome of Bob and Monica, Don and Rhonda returned to Durango.  We went right by the Strater as it turned out, and on to the Kennebec Bakery and Cafe in nearby Hesperus.  Exemplary meals were had by all, with plans for another foray into the great Durango outdoors, where I'm sure we'll take the measure of some pinon pines of note...
It's the middle of the night, and with not sugar plums but southwestern white pines dancing in my head...trying to work through a tree measurement problem, that I'm sure Bob will conjer up a solution for with the bat of an eye.

Here it is, without any graphics that I can bring to bear, and only my words to describe (we're in trouble now!):
Given: A leaning conifer tree on flat ground (need to make this simple).
Definition of lean? That offset measured in a horizontal plane, between the top and the bottom.
Given: This tree's lean is all to the North (0 or 360)...in the amount of 10 feet (in that horizontal plane oriented NS/EW).
Given:  Bob is standing due West (270) of this tree, and he measures the top at +134.9 and the bottom at -4.1, with a hypsometer and a laser rangefinder. The tree is 139.0' tall (plus or minus 0.4).  
Given: Bob stands due South (180) of this tree, and his top and bottom readings totalled to 139.4.  He conservatively rounds down to 139.0 and records tree height as such.  Day done?

Problem: Don stands to the Northwest of same tree (315), and extracting every bit of accuracy he can from his lowly Nikon 440, measures top and bottom, and records a tree height of 139.2, and upon hearing Bob's results, is satisfied that they have nailed that tree, heightwise.
What's the problem?  One of the reasons we are out here is to gather data on how far off of vertical that Colorado species of trees in this region are.  The problem is that we don't know how far off vertical these trees are (outside of my given above) as our sine/sine method is measuring the vertical plane.

My elementary geometry fails me at this point.  As a long time 'tangent' height measurer, the best I could do was circle around the tree until I found a location that 'took all the lean out', or in the case of the mythical tree Given above, I would either measure it from 0 degrees North of the tree or 180 South of the tree.  So in my head, having been a surveyor, I see the solution in a transit or theodolite, or these days, an "All-Station", where I can position the instrument where I perceive the maximum offset (270) and measure the angle describing the two points where the top and bottom cut through the horizontal plane (which would with superior geometric skills turn out to be 10'.
We don't usually cary transits or theodolites or All-Stations several miles into the woods, without consideration.

This is where I'm sure Bob, who sees the three-dimensional world in sides and angles and proofs and theorems and will be able to relate all these together and say_____________________!

Right now I'm guessing that it could involve a tangent measurement....what do you think, fellow ENTS/WNTS?  What are your ideas?  We'll check back tomorrow afternoon/evening and see what solutions emerged!