Summary:
After Techs that increase your troop buffs (from the Military, Defense, and Military Advance Trees) the Medical Aid Tree is going to become one of the most important sets of Techs for hardcore PvPers. You will find a wide variety of Techs that will improve your quality of life by making PvP a less costly venture. Even if you don't participate in PvP you will still want several of these techs to help you heal faster after running high level bosses.
Essential Techs
Hospital Scale and Super Hospital Scale:
These are pretty self explanatory; one increases your Hospital Capacity by a straight number value, while the other by a percentage. Maxing out your Hospital Capacity is vital as it will prevent you from having to rebuild (or revive during SvS) as many troops after battles.
Light of Server War and Adv Light of Server War:
Without having these Techs maxed for your Academy Level you might as well sit out for SvS. These Techs play the very important role of increasing the total power amount of troops that you can revive in a single go when using Source of Life (Sol) during or right after SvS (you have 3 days to revive troops after SvS ends or else they disappear forever.
Ground, Mounted and Ranged Troop Healing:
These Techs increase your healing speeds for their respective troop types. Maxing these will greatly reduce the number of Healing Speedups you will use over time. This may not seem like a big deal to someone who doesn't yet participate in PvP, but it will when you inevitably make the plunge.
Junior, Medium, Senior, and Super Surgery:
Like the previously mentioned group of Techs, these increase your healing speed as well. This time instead of Troop Type based boosts, the boost is applied by blocks of Troop Tiers. You definitely want to keep these Techs as high as possible to help further reduce the strain on your Healing Speedup stockpile.
*Disclaimer: These remaining Techs are absolutely essential if you plan on doing any sort of Attacking. If you are a Defensive player and let the enemy come to you, they basically do nothing for you and can be put on the back burner until you decide to become the hitter.*
First Aid and Super First Aid:
These two Techs maxed give you a 10% Death-to-Wounded rate when you are attacking. This means that when you attack someone, instead of dying outright, 10% of what would have been dead troops instead go to your Hospital to be healed. This will helps a ton outside of SvS when you would normally be losing that 10% outright and helps reduce the number of Source of Life (Sol) you will use during SvS.
Ground, Mounted, and Range Troop Rescue:
On top of the First Aid Techs, you can also continue to increase your Death-to-Wounded rate for each troop type even further.
Note: If your hospital is full, dead troops cannot and will not be changed to wounded as you cannot exceed your Hospital Capacity.
Troop Revival and Super Troop Revival (all Troop Types):
These Techs do one thing; Bring a very small percentage of your troops back to life immediately after the battle has ended. While the percentage seems miniscule, it does make a difference and helps lower your overall Healing Costs.
Non-Essential Techs
Military Resource:
I only put this as a Non-Essential because you should try to go into SvS with as little resources open as possible. Increasing your warehouse protection power during SvS is rather trivial compared to the rest of the awesome Techs in this tree.