Okay I may have found a type of solution for 145.

145 = (12)^2 + 1

= (3^2 + 3)^2 + 1

= 3^4 + 2*3^3 + 3^2 + 1

= 81 + 54 + 9 + 1

= (81 + 9) + (54 + 1)

= (90 + 55)

GCD = 5.

or = (81 + 54) + (9+1)

= (135 + 10)

GCD = 5.

If we keep the terms separate maybe we can get it to work. Moreover I've looked at a bunch of trees and I think we could terminate the trees at either 1, 2 or 3.