Colin MacDougall pleaded guilty to four counts after the incident in August 2010 at Manor House Wildlife Park, Pembrokeshire.
Gruff Davies-Hughes, three, was injured, and his mother was also hurt.
MacDougall will be sentenced at a later date, Swansea Crown Court heard.
The prosecution decided it was not in the public interest to proceed with the case against Ryder Richardson because she had no involvement with tree management on the park in St Florence near Tenby.
The court heard that MacDougall was "more responsible" for the safety and upkeep of the hundreds of trees at the zoo.
David Morgan, prosecuting, said: "A three-year-old boy and his mother were injured as a consequence of a tree falling.
"There is evidence to link a breach of health and safety laws with that incident. It is unclear whether a proper regime would have prevented it.
"Anna Ryder Richardson's day-to-day involvement with this part of the company was far less significant than MacDougall's."
Ryder Richardson had denied breaching the Health and Safety at Work Act 1974 but her husband pleaded guilty to four charges - two in his own name and two in the company Manor House Wildlife Ltd.
Ben Compton, defending Ryder Richardson, said: "There is no evidence at all to link her with any form of tree management at this wildlife park.
The park was investigated by Pembrokeshire council who decided to prosecute the couple.
MacDougall, who had previously pleaded not guilty, will be sentenced at a later hearing.