There are some trees around with branchy tops that seem wider than half of 10M, maybe even half of 20M, and above the nearby rooftops. Can a wire dipole or a vertical be mounted inside a tree? If so, what kind of problems might such an antenna present? And, even if functional, would there be any benefit to putting an antenna inside a tree, versus a ground mounted antenna below the rooftops and treetops.
Yes, you can mount an HF antenna in the tree tops. I'd wager there are many many wire antennas strung through the trees around the world, two in my backyard.
I imagine there is some effect, probably a greater one when there are leaves on the tress then now. But, I have had plenty of stateside and DX contacts with this arrange, so it does work fine. Now, I am sure it will work somewhat better in free space, but I don't have a choice.
Generally speaking higher is better for antennas. In the trees there is less tree to go through to get to the sky than beneath them, so I am pretty sure there is little advantage to mounting a dipole beneath the canopy of the trees. Although I have seen reference to mounting dipoles low to the ground in ECOMM situations where NVIS is desired.
Verticals will depend on the type and what they use for the image antenna.
There are bits of anecdotal evidence on the internet that the end of a dipole can set fire to a tree. This is because the voltage at the end of the antenna is very high (compared to the feed-point voltage). Aggravating conditions include using a linear amplifier (increasing the voltage) and/or operating in rainy conditions (compromising the insulation).
One important aspect of antennas in close proximity of trees or even touching branches is problem with correct and persistent tuning of such antennas. Needles to say totally unknown characteristic of radiation. Correctly calculated and tuned length of wires are often no longer correct in trees. SWR is something magically changing or flat across whole band without abilities to lower it down in any frequency etc. Generally speaking antennas going between trees and branches it is bad idea and may bring a lot of frustration during setting up an antenna. Trees as support isolated from antenna yes that works quite well.
If you do put your antennas in or between trees, make sure no uninsulated part of the antenna touches the tree, especially when wet. Because if it does, the tree becomes part of the antenna and will detune it. (Catching on fire or worrying your tree might get radiation burns is a separate issue.)
But even if the tree doesn't detune it, it may absorb some of the energy and attenuate it, especially if the antenna is parallel to parts of the tree, or you are doing vhf or uhf where the leaves absorb even more.
I also have my HF antenna hanging from a tree. But I also use an antenna tuner with a wide band antenna, and the antenna is pretty far out from the trunk.