The macaw has a brain the size of an unshelled walnut, while the macaque monkey has a brain about the size of a lemon, according to researchers from Vanderbilt University in the US.
However, the macaw has more neurons in its forebrain - the portion of the brain associated with intelligent behaviour - than the macaque, they said.
Researchers measured the number of neurons in the brains of more than two dozen species of birds ranging in size from the tiny zebra finch to the six-foot-tall emu, and found that they consistently have more neurons packed into their small brains than are stuffed into mammalian or even primate brains of the same mass.
The study provides a straightforward answer to a puzzle that comparative neuroanatomists have been wrestling with for more than a decade - how can birds with their small brains perform complicated cognitive behaviours.
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The puzzle was created by a series of studies beginning in the previous decade that directly compared the cognitive abilities of parrots and crows with those of primates.
The studies found that the birds could manufacture and use tools, use insight to solve problems, make inferences about cause-effect relationships, recognise themselves in a mirror and plan for future needs, among other cognitive skills previously considered the exclusive domain of primates.
"We found that birds, especially songbirds and parrots, have surprisingly large numbers of neurons in their pallium: the part of the brain that corresponds to the cerebral cortex, which supports higher cognition functions such as planning for the future or finding patterns," said Herculano-Houzel.
"That explains why they exhibit levels of cognition at least as complex as primates," she said.
That is possible because the neurons in avian brains are much smaller and more densely packed than those in mammalian brains, the study found.
Not only are neurons packed into the brains of parrots and crows at a much higher density than in primate brains, but the proportion of neurons in the forebrain is also significantly higher, researchers said.
The findings were published in the journal Proceedings of the National Academy of Sciences.