Just to be sure you are aware, the HDHR3-US has no analog capability. However, I own both the new HDHR3-US and the earlier HDHR-dual:
http://forum.tvfool.com/showthread.php?t=820 and I'm glad I do. A HTPC can support several HDHR tuners as well as other brands with analog capability. So if you are planning to watch both analog and digital signals, you may need to consider products from Hauppauge and their competitors.
The tree sounds like it could pose a problem but I think you would be able to get the majority of the signals to the SW using a Winegard HD7698P or the combination of an Antennas Direct 91XG and Antennacraft Y10713. With either option, an Antennas Direct CPA-19 preamp would be a good choice.
Looking SE, I'd use a separate antenna or combination. At a minimum a 4-bay such as the CM-4221, U4000, HD4400, DB-4 or DB4e. If CFPL is still on the air on CH-10 and you want to see it, another Y10713 dedicated to that signal would be in order. I'd try the SE facing antenna(s) with no preamp... another CPA-19 could be added if it proved necessary.