I am curious about something, how exactly do bound weapons work? Do the relative weapon trees boost the damage of their bound version (ie, does the barbarian perk boost the damage of the bound battleaxe)?

yup

are you sure? I dont want to waste perks on making bound weapons a primary fighting style if it is mainly just meant as a back up style

I am curious about something, how exactly do bound weapons work? Do the relative weapon trees boost the damage of their bound version (ie, does the barbarian perk boost the damage of the bound battleaxe)?

Yes:
-Bound Sword is a one-handed sword, for all purposes.
Damage: 9 (equal to unimproved Steel)
Mystic Binding: 14 (equal to unimproved Daedric)

-Bound Battleaxe is a two-handed axe, for all purposes.
Damage: 17 (weaker than unimproved Steel; equal to unimproved Steel Greatsword)
Mystic Binding: 22 (equal to unimproved Glass)

-Bound Bow is a bow, for all purposes.
Damage: 18 (42) (slightly weaker than unimproved Daedric using Daedric arrows)
Mystic Binding: 24 (48) (equal to an Exquisite Daedric bow using Daedric arrows, marginally inferior to a Legendary(1) Daedric bow using Ebony arrows)

Note that all Bound weapons are weightless. For sword and axe, this means the minimum possible Stamina drain for power attacks.

Edit: Damage values are base. They are affected by the skills, the x/5 root perks, and appropriate enchants and potions.

Thank you, much appreciated