The question is: will the tree pull the car out for you?

The question is why wouldn't the tree collapse in the process.

If d=0 then also F=0, so T can have any value.

I have no idea what you mean here.

How is this not asking about underlying physics concepts, am I missing something? How is this a closed question?

As for what @my2cts was talking about, he means that $\theta$ is a consequence of $F$, and thus, $d=0$ if and only if $F=0$. So in reality $T=0/0$.

@Conreu I asked the proof about the advantage in force and so the question was closed. So I asked a new question (after proving it myself) which is entirely different and which was again closed as someone thought the question was same. So I edited this question.

@my2cts that's fine but as soon as the force has been applied, and the string starts to form the V, the force will be finite but d would tend to zero.

'the force will be finite but d would tend to zero' You can't fool nature. F=0 implies d=0 and vice versa.

@my2cts, The only thing a winch can do is pull two objects toward each other. One end of the winch line is made fast to the car, but if you want the car to move, then the other end must be made fast to something that will not move, even when there's enough tension in the line to move the car. Now, what thing shall we use as our anchor? Ooh! See that tree over there? The one that you wanted to forget...?

Re, "why wouldn't the tree collapse?" TLDR: When you pull sideways on the midpoint of the rope, the rope will stretch. (I guarantee it!) And, if you pull hard enough, something will move. Either one of those things will allow the rope to bend where you pull on it, and that will limit the tension to some finite value. Also, if the tree is rooted in the Earth more strongly than the car is stuck in the mud, then if and when something moves, the thing that moves will be the car.

@SolomonSlow Obviously the winch should be mounted on a truck, not on a tree. Winches don't grow on trees.

Would you mind including MathJax or plain text transcription of the calculations? Despite having 20/20 vision, I severely struggle to decipher anything useful from that microscopic image that features your proof.

@SolomonSlow ‘The only thing a winch can do is pull two objects toward each other. ’ That is more than what the tree can do, hence I suggest a winch instead of a tree. Professional car towers will agree.

@my2cts, This is only just a guess on my part, but I don't think that the OP actually is trying to get a car unstuck. But, I could be wrong. In which case, your idea is a good one. They should just call for a tow truck. Really though, I think that the actual reason for their question is that they are trying to understand the meaning of the infinity that appears in their mathematical model of a vector pull.

P.S., Vector pull is a technique that I actually have used to get a boat un-pinned on a lively little whitewater creek. We don't carry electric winches, nor the batteries to run them in whitewater kayaks. And, getting a tow truck to the scene (deep in a narrow ravine, surrounded by dense woods, more than a mile from the nearest road) would have been absolutely impossible. We do carry long ropes though for just such occasions, and the vector pull (if you keep pulling, and then re-setting it, pulling again, and re-setting it again, etc. until the boat comes free) can save the day.

@SolomonSlow I cleared the mystery of the infinity in my second comment. Since that did not seem to be solve the issue it seems like the OP is actually stuck in the mud. Hence my advise to call in a towtruck and avoid unnecessary damage to the tree.