The cat is climbing (up) the tree.

[Phaedrus]

Those sentences don't sound too strange to me.

They don't to me, either, to tell you the truth. I just wanted to believe they did! Oh well, I suppose it is a useful result to realize that climb up the tree can be parsed in two ways -- as an intransitive verb adverbially modified by a prepositional phrase, or as an inseparable transitive phrasal verb (climb up) followed by a direct object -- and that it is an atelic VP on the former interpretation and a telic VP on the latter.

The mind of the native speaker will naturally go with the parsing that makes sense with a normal meaning. Thus, if we hear It took the cat thirty seconds to climb up the tree, we naturally assume the transitive phrasal-verb meaning, that the cat took thirty seconds to make its ascent to the top -- not that it took the cat thirty seconds to begin the activity of upward tree-climbing (intransitive verb + PP).

But contrast:

? The cat completed/finished climbing along the bough of the tree.
? It took the cat thirty seconds to climb along the bough of the tree.

 

[Tarheel]

Say:

I think both sentences are okay. (No "the'.)