'Game of Thrones' season 8 episode 3 saw the Night King destroyed like he was created, under a weirwood tree and by a girl
It would not be wrong to say Bran chose to wait for him beneath a weirwood tree because he saw the Night King being transformed from a man to a zombie near one
Nothing in George R. R. Martin's story remains in the background just for the sake of it. Throughout the series of 'Game of Thrones' the author, and later the creators of the HBO show - David Benioff and D. B. Weiss - have diligently followed Chekhov's Gun theory where nothing they have mentioned earlier goes unnoticed. Similarly, episode 3, which finally saw the Great War between the dead and the living come to an end, wrapped up almost the entire saga of the Night King.
While on one hand, the Night King's death gave us the unexpected reality of Azor Ahai, on the other hand, his death was so very ordinary you cannot help but sympathize with the Night King. Bran had already stated the Night King would come looking for him in order to erase the last living memory of humankind.
However, when the Night King finally met Bran and was about to take out his sword to kill him, Arya sneaked in from behind and stabbed the Night King with a Valeryian steel knife right in his chest - exactly where the man was stabbed thousands of years ago by the Children of the Forest which turned him into living dead. Also, we should not fail to consider how the Night King died just where he was born. In Bran's vision back in Season 6 we had witnessed the Children of the Forest gather around the weirwood tree to turn the man into the Night King.
The tree was surrounded by stone structures, giving a perception of a spiraling pattern which almost became a recurring phenomenon in the show. Each time the Night King was mentioned we would see similar patterns - sometimes with the distorted arms and legs of the Wildlings and the final time with Ned Umber hanging right in the middle of the pattern, a sword stuck to his heart. While most of us perceived the pattern as the Night King's unmentioned relationship with the Targaryen's, it turned out that all the while the Night King was actually hinting towards the place he was born...the weirwood tree amidst the spiraling stones.
It would not be wrong to say Bran chose to wait for him beneath a weirwood tree because he saw the Night King being transformed from a man to a zombie near one. On other occasions, Bran has been extremely close to the Night King and was touched by him. Hence, perhaps throughout it was Bran's idea to lure the Night King into the exact place where he was created and let him die right there. This is where the Night King's story comes to an end, and let's just say it certainly makes just the perfect circle of life, even for a man who has been dead for thousands of years.