If there was a planet half the earth’s diameter, for which every other aspect of the earth was changed proportionally to keep the Earth’s perfect ratio intact (including having the perfect orbital distance from the sun, etc), would animal and plant life, in turn, grow proportionally smaller? Would an Oak tree grow to be 80 feet, or 40?
Assume the earth is positioned in just the right place in space with just the right diameter/mass/surface etc to make all this possible.
Half the planet diameter means 1/8 of the mass (mass goes like volume, thus $(1/2)^3=1/8$, but, since the surface will be twice closer to the center of mass it will also be a 4 times stronger gravity. In total gravity will be half of what we have on Earth.
The escape velocity of such a body would be half of what we have on Earth, and based on this never enough praised chart
we see that the planet would be able to retain oxygen and carbon dioxide, but not water. Sadly, the only life we know is based on water. So, your planet, with the feature you express, could likely not be capable of hosting life as we know it.
Actually no - there is a good chance that the plant and animal life could even be bigger!
We see this effect on isolated islands on Earth - where the smaller landmass can result in gigantism in certain species.
If you read that and apply it to your question...it's tricky. I would expect some species to grow bigger (eg birds - especially considering lower gravity would make it easier to fly), and some to grow smaller (eg bison/deer/whales because there's just less grazing space for maintaining a sufficiently large breeding population unless they shrink - see insular dwarfism for some real-world examples).
As for "what would happen if you placed a human in this world?" - There is an excellent answer Human/mammal growth on Mars on what would happen to a human on Mars, which is just over half the size of Earth.
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Commented
Jul 11, 2018 at 8:40
This started as a comment on another answer but got too large, apologies if it reads like one.
Mars (diameter 6790 kilometers) is only slightly more than half the size of Earth (diameter 12750 kilometers). so for ease of explanation, I'm going to use it as the example despite being BIGGER than the planet in the question.
Atmosphere
One of Mar's biggest issues as a life-sustaining planet is that its core was too small to stay molten enough to generate a powerful magnetic field, (this was indeed compounded by its extra distance from the sun). Once that magnetic field diminished in strength then the Solar wind was able to rip Mars's atmosphere off making the planet unsuitable for complex life (we aren't ruling out microbes etc yet) so that's a big problem for life evolving at all, let alone smaller, bigger or the same size.
Gravity
Since Mars is a lot smaller it has less mass than Earth, the surface gravity on Mars is also a lot less than the surface gravity on Earth. The surface gravity on Mars is only about 38% of the surface gravity on Earth, so if you weigh 100 pounds on Earth, you would weigh only 38 pounds on Mars. this then causes the issue of atmospheric density... if you managed to have the core stay molten long enough for complex life to evolve, you'd still have an atmosphere a lot less dense than Earths, meaning any life would need to be able to survive on significantly less air than they do on earth.
Sorry to be so negative but the size of life doesn't really come into it when the likelihood of life at all is so very small. if you like Handwavium as the explanation of how life could survive, then I'd recommend mixmastered's answer to the most likely outcome on what that life would be like.
Not to detract from the other very fine answers, but I suspect that there are variables you could modify. For instance, if the composition of the core were different to the Earth's then you might have a sufficient magnetic field and increased density. This might allow a smaller planet to retain atmosphere and have a similar surface gravity to Earth.
For the sake of argument, let's say that the planet is half the size of the Earth but the other factors are broadly similar - the atmosphere, gravity, temperatures, length of day, etc.
The question then is whether the reduced space alone would result in evolutionary selection pressure towards smaller creatures.
Personally I think that it's unlikely. Smaller populations, maybe. But I don't see any particular impediment to 'normal' (Earth-normal that is) sized creatures being the norm. The balance we have reached seems more to do with the direct factors: gravity, atmosphere, temperature, etc. Indirect factors like room to expand and so on would seem to be more likely to result in smaller populations or reduced diversity than to produce smaller individuals.
If you were to find a rock like this, Terraform it, populate it with an Earth-native ecosystem and let it develop independently for a few hundred thousand years and things probably wouldn't change all that much more than on a world more like Earth in size. Maybe a little less, given the reduced opportunity for geographic isolation.
Nope, the (plant) life would likely be bigger than on Earth.
Assuming the density is the same, the mass would be 1/8 that of Earth, and the radius would be 1/2 that of Earth, so the surface gravity would be 1/2 that of Earth (proportional to M/R^2).
This would mean that an oak tree would grow to 160 meters instead of 80.
However, the atmospheric pressure would be much lower and effect for the partial pressure of oxygen would be more pronounced still (even if offset by the larger plant life). This means that the animal life would be smaller than on Earth, most likely, but definitely not just twice as small.
