However, other neighbours of the couple say the tree is a local landmark and are calling for it to stay.
"This tree has been part of my family's life since 1966 when we moved. It has been a feature of our children's growing up, always there as a reminder of the beauty of nature in a dense, central urban environment," writes Colin Jacobson in an official objection to the plan.
"I consider the proposal to fell this tree as being a drastic overreaction to the alleged problems."
In a report to the council, subsidence expert Stuart Harris says that the cracks are clearly caused by the tree.
"In order to stabilise the property and prevent further damage occurring in the future, the cause of the movement needs to be addressed," he says.
Camden Council is now considering the application.
Camden is chock-full of the rich and famous, and they're not slow to defend their properties. Last September, for example, Great British Bake Off star Sue Perkins successfully prevented a nearby pensioner couple from building a bungalow at the bottom of their garden.
In the same month, Sherlock star Benedict Cumberbatch was finally given permission to add a new extension and boiler room to his Camden home after neighbours kicked up a fuss.
But one of the most amusing battles between celebrity neighbours took place in nearby Holland Park. After Robbie Williams bought the former home of Michael Winner, he submitted plans for extensive alterations, which were approved.
But the work rumbled noisily on for months, infuriating neighbour Jimmy Page, who complained - unsuccessfully - that it should be stopped. Williams has three years to complete the work, the council said.
Highest-earning dead celebrities
Highest-earning dead celebrities
MJ may have died over six years ago but his legacy still lives on. The King of Pop made $115m this year, but how? Cirque du Soleil in Vegas showcasing Michael Jackson One is one of the source's of Michael Jackson's cash flow but the Mijac Music catalogue and music sales also contribute.
Elvis pulls in most of his money from ticket sales at his Graceland home. Now it may seem surprising that this rock and roll legend could generate that much income from tours of his home, but it makes more sense when you realise a regular adult ticket will set you back $77!
The Marley Beverage Company and House of Marley are just two of the companies bringing in the big bucks for reggae legend Bob Marley. Music sales are also obviously a huge contributor with more than 75 million albums sold so far.
Celebrity fragrances might not really be that sought after any more but White Diamonds, one of Elizabeth Taylor's perfumes, was first released 24 years ago and continues to be sold to this day proving that maybe there really are some things that never go out of fashion.
Marilyn Monroe is undoubtedly still an international fashion and film icon. In 2011, Authentic Brand Group bought the rights to her brand and it's thanks to them that she's still pulling in top dollar as well as a clothing line at American store, Macy's, and a range of lingerie.
John Lennon was part of one of the biggest musical acts of all time, The Beatles, although unfortunately his stardom was cut short when he was shot dead in 1980. The songwriter still gets his share of the album sales though and is reported to take $12 million each year.
Einstein is one of the most famous scientists in history having come up with the general theory of relativity. Branded mechandise and memorabilia is the main source of the theoretical physicists revenue now though.
Paul Walker was just 40-years-old when he died in 2013 in a car crash, but his legacy still lives on thanks to the film franchise, The Fast and the Furious. The most recent film took $1.5 billion and although Walker died during filming, he still appears in the flick.
Bettie Page was one of America's iconic pin-up girls, even gaining the nickname 'Queen of Pinups' thanks to her trademark look. The model was also one of the earliest Playmates of the Month to feature in Playboy magazine. Page now makes most of her money at the hands of fashion, clothing and lingerie lines.
Theodor Geisel was responsible for some of the quirkiest and best-loved cartoon characters and children's book heroes. The artist and author published his works under the title of Dr.Seuss with some of his most famous creations including The Lorax, The Grinch and The Cat in the Hat still netting him money to this day.
Not to be confused with the award-winning film director of the same name, Steve McQueen was, at one point, one of the highest paid film stars in the world. Some of McQueen's most famous films include The Thomas Crown Affair, The Magnificent Seven and The Great Escape. His image is still used in adverts and commercials, bringing in $9 million worth of revenue.
James Dean was a certified heartthrob and cultural icon of the 1950s who sadly died in a car accident at the age of just 24. His iconic image has been used throughout the last 60 years in a variety of ways all of which have been bringing in the cash. Most recently Dean's image was featured on a designer Dolce & Gabbana tshirt.