I'm not sure I understand why the tree had to be climbed to be measured. Isn't that what trigonometry is for?

 Yes, and no. Trigonometry works if trees grow with a habit which matches the assumptions of a right triangle.Before laser rangefinders were cheap and accessible, the method used (angle and an approximation of horizontal distance) would often grossly overestimate the height of the tree.The modern sine method is much better, but still had problems with leaning trees, and will tend to underestimate height.The most reliable method is to get someone up there and do a tape drop.
 Seems like a great job for a drone.
 You could use a theodolite. But remember that the ground is not flat. So you need to measure the vertical difference in height from the base of the tree to the theodolite. And you need to measure the horizontal distance, which would require other measurements in other locations. And you need a location where you can clearly see the top of the tree. This is all a bit complicated for someone without topographic surveying experience.A tape measure is simple and easy to do accurately. And of course they will want to climb the tree anyway.

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