Just like the Grey Dwarves and Greylings, the Elder (their king i guess??) is made of wood and therefore burns quite well if you use fire arrows against him. The pillars around the summoning area are great for avoiding his "ranged" attacks, and when he summons the roots, simply run to a different pillar, duck behind and shoot him with fire arrows before he shoots the vines at you again.... repeat... Pretty easy boss.

Just for general information... I am not telling anyone how to play, but Bronze armor is somewhat a waste of time and materials (at least right now), as you can survive just as well (or better) using upgraded troll armor, which has no movement penalty and therefore doesn't drain stamina nearly as fast as bronze armor does.
I often use Troll armor throughout the game until after completing the swamp biome, and have no problems running across the Plains biome wearing troll armor, if I'm just passing through... It's pretty good armor overall.

The Bronze ingots are better spent on bronze tools and weapons, and I DO understand the tendency for people to craft all the bronze armor, as it actually LOOKS like your first "real" armor, but just pointing out that it's very possible to survive just fine with upgraded troll, especially since you can move / run / evade mobs much easier without losing as much stamina. MANY people skip the whole Bronze armor thing as it really isn't worth the time or effort.