"What is the height of that tree?"

Translation: Chiều cao của cái cây đó là bao nhiêu?

[JCMcGee]

Wow...weird that it's cái cây and not cây cây

[Huy_Ngo]

Yes, but you can think of it like this: tree is a (direct) subclass of object, the classifier for object is "cái", therefore "cái cây".

[TehVanarch]

cái is the generic classifier/counting word and functions like the definite article "the" here, singling out a certain object. It would be redundant to say tree twice.

[JCMcGee]

Yes...but can "cây" not also be used as a classifier?

[TehVanarch]

In this case it would be redundant, you would use "cây" with a stick-like object or a type of tree.

e.g.1 Tôi muốn mua một cây thông = I would like to buy a pine tree.

e.g.2 Ông kia cần một cây gậy mới = That (old) man needs a new walking-stick/cane.