Forensics expert Claire Bolton died after the 50ft ash smashed into her green Subaru Impreza as she drove down a country road near Hawkhurst, East Sussex, in January this year.
The 26-year-old, from Staplehurst in Kent, had no time to swerve and was killed instantly when one of the large branches broke through her windscreen, the hearing was told.
The tragedy occurred after heavy rains caused havoc across southern England earlier this year, the inquest at Hasting Magistrates' Court heard.
Giving evidence, Pc Mark Hill told how near the tree had fallen to Miss Bolton's son, who was strapped into the back of the car.
"The tree got to within an inch of the baby," he said. "It was extremely close."
Witness Keri Havens said the tree simply toppled, leaving Miss Bolton no time to get out of the way. "It was gusty and there was a bit of rain," she said. "The tree just came down smack on top of her car and made it bounce.
"The tree pushed it to the other side of the road. It all happened so fast the lady driving didn't even have time to break."
The inquest heard how the tree, which was on private property, was deemed healthy by tree surgeon Mark Cruttenden.
But forestry officer Andy Piper said the fall was caused by a build up of water surrounding the roots and heavy winds battering the tree.
Mr Piper, who works for East Sussex County Council carrying out tree inspections, said there was no way anyone could have predicted the tree would fall when it did, following heavy showers.
"The roots were waterlogged and with the gusting winds from the day before that had just come over, it would have been the wind that caused it," he said.
Recording a verdict of accidental death, coroner Alan Craze said: "It is the view I take that she had absolutely no chance.
"This has been described by witnesses as a freak accident. Nobody could have predicted that the tree would fall at the precise moment that Miss Bolton would have no chance of avoiding it.
"Thank God the baby was miraculously unharmed."
Miss Bolton's fiance and other members of her family fought back tears as details of her death emerged. Distraught, they refused to comment after the hearing.
A family friend said: "They still cannot come to terms with what happened. It was an absolute tragedy."
Kent Police, who employed Miss Bolton, paid tribute to the dead officer.
Detective Chief Inspector Colin Croucher, former head of the forensic department, said: "Claire was an extremely valued and popular member of the forensic department where she worked as a performance analyst and she will be very sadly missed by all her colleagues.
"Prior to working in the forensic investigation department, Claire was a processing and administrative assistant in Ashford.
"All those who knew and worked with Claire are deeply distressed upon hearing the tragic news of her death and our thoughts remain with Claire's family."