Check whatever building codes or electrical code that you use in your country or area. I know a lot of people feel it’s better not to ground because they feel it attracts lighting but the bottom line is to follow the codes. The last thing you want is a lightning strike and fire and then have your insurance company argue in court that you where negligent with your installation and you were partially responsible for the damages.
They might get out of paying the entire cost of repair.
In the US the National Electrical Code would require grounding a mast if it was for an antenna but I think your mast would have the same attraction to lightning as an antenna. I have my anemometer mounted 10’ above the roof on a chimney bracket with a heavy ground wire to a ground rod. The ground should also be connected to any other ground system you have, such as your electric service, unless that isn’t permitted in your country.
As far as other things acting as lightning rods and protecting your house, the general rule of thumb I’ve learned from lightning protection contractors is you’re in a zone of protection if you’re within an area less than the height of the grounded item you’re depending on for protection such as a building that has lighting protection installed on it. I don’t think trees would be good enough conductors to count on for protection.