You must have the word "is," because the clause "how old a tree is" is a noun-phrase. In this case, it takes the position of the subject. The entire phrase "how old a tree is" should be thought of as a single unit. If you wanted to rephrase it, you could simple say "The age of a tree," but in the former phrase, you cannot remove the verb "is." Without the verb in that case, it's not a complete thought.