The notion "transitive" is not entirely clearcut. One definition I've seen is that a transitive verb is one that requires a following direct object to complete its meaning. (At least, it follows ordinarily.) But this is a grammatical classification, not a notional one.
Let's take your example "I attempt to climb the tree" and compare "attempt" with "try". Answering "Will you attempt to climb the tree?", I could say, using a direct object:
I will attempt it.
I will try it.
but, not using a direct object, there is a difference between these verbs:
*I will attempt.
I will try.
Evidently there is a difference between the verbs: "attempt" requires an object to complete its meaning, but "try" does not. In the way I would usually use the term "transitive", this is a difference in transitivity. "Attempt" is transitive, but "try" is not.
In your example, I would say that "to climb a tree" is a noun phrase and is a direct object. It satisfies tests for being a noun phrase, e.g. it can be pronominalized, and it can even be passivized: "To climb a tree has often been attempted."
Wikipedia has an article on Transitivity, which was first elevated to an interesting topic in syntactic theory by Sandra Thompson and Paul Hopper.