The garden includes a dell demonstrating the evolution of land plants - one of the first representations of plant classification based on DNA gene sequencing.
Bellamy said: "These paths really touch history and that's so important. If we can't touch history and learn from history, then our knowledge is very trivial. This place will give an awful lot of people a lot of hope."
Garden director Professor Simon Hiscock added: "The new garden is unique among UK botanic gardens in telling the amazing story of plant evolution, from the lowly liverworts to glorious flowering plants like the giant Amazon water lily."
Garden curator Nick Wray said: "The garden will be here for many years to inspire and educate the people of Bristol for generations to come."