Please Explain Kotov's Analysis Tree Method

I would just make on other point.  When you have a forced line that goes deep with only a few branches along the way (what Kotov would call a 'bare trunk with branches'), you should note but not get detracted by the branches, and continue to follow the most forcing line all the way to checkmate (or some conclusive ending).

Then you backup to the next closest branch, and see if your opponent had a better reply.  If all of those replies lead to the same (or worse) outcome as the mainline, then backup again to the next closest branch.  If you find your opponent has an alternative reply that saves him, then look at your own candidate moves at that same position to see if you have better.

You keep working your way down the tree until you have carefully checked all branches.  Make sense?  Sort of like this:

1. W+ B (only) 2. W+ B (or 2...Ba, 2...Bb) 3. W+ B (only) 4. W+ B (or 4...Ba, 4...Bb, 4....Bc) 5. W+ B (only) 6. W#

So this would be a 'bare trunk with branches' since visually it would look like a long trunk with only two branches.  We now know white can force mate if black chooses these moves.  So we backup one branch at a time to see if black can avoid checkmate, starting with black's alternatives at move 4.  We then calculate each option and see what we come up with after 4....Ba, 4....Bb, and 4....Bc.  

Let's assume they all lead to the same outcome as the mainline - checkmate.  So then we backup again to move 2 and see if black can escape checkmate with 2...Ba or 2...Bb.  Let's say 2...Ba leads to checkmate in 1, but 2...Bb we follow for many moves and find that it allows black to escape checkmate (or maybe even allows black to checkmate white by force!).

We then take another look at white's move 2 and see if he has alternative moves.  

Hope that makes sense.

Whoops, didn't see someone had posted the winning line already. (Black either gets mated or loses a piece.)

So, in conclusion, if I ask myself what is the correct way to calculate in chess, should my answer be-

1)Look at all candidate moves.

2) The candidate moves should be-1)Threat 2)Checks 3)Capture

3)While analysing, calculate the most forcing line, then the next more forcing line etc. 

4) If you do not get a tactic, look at candidate moves which improve your position.

Doirse, or anybody else, can you explain what precisely do we mean by a threat?

well sort of...

- when attacking, you should develop candidate moves by visually searching for specific clues that might indicate a tactic (eg, alignment of pieces, knight proximity, undefended piece, exposed king, etc).  Those visual clues should tip you off that a tactic might be available.  You can train your ability to spot these clues, and the multiple ways to exploit them, by drilling lots of tactics.  You should calculate as many of these candidates as you can find, time permitting, and starting with the most forcing move often helps move through the variations quickly.

- when defending, you only need to consider the five defensive ideas.  You should calculate all of them.

- if there are no tactics, play chess!

- threats:  anything bad your opponent can do to you.  Some are obvious, like a threat to capture a piece, execute a tactic, or deliver checkmate, while others are more strategic like a threat to trade down to a won endgame, gain space, etc.  

But don't forget Cecil Purdy's axiom about threats -- "you have to recognize the unreality of the opponent's unreal threats".  If you think you see a threat, it might not be.  Take the following for example -- white just played 1. Ne6 attacking the queen and threatening checkmate...or is he?

I solved the puzzle at the first attempt.

Well, thank you very much Doirse. When I started this forum I was in the early 1400's and now I am closing in 1600! 

I have found the golden approach to chess.

1) Find all your opponent's probable tactics

2)If there are any, defend. If there aren't any:

3)Look for all your forcing moves to find any tactic (follow Doirse's method ). If you find, execute it. If you can't find any, play basic but solid positional chess. 

About positional chess, I read here: http://www.mark-weeks.com/aboutcom/ble26pos.htm

It may not be the best approach, but it works wonders for me. By the next two months, I hope to cross the 1800 mark (I am now very close to 1600)

Golden approach... That's a good one.

It's good you have enthusiasm.  Try not to be too discouraged if it takes longer than 6 months to become a grandmaster. (At 200 points a month, you can be at 2800 in only six more months!)