Thank you for your patience.

The decision tree has parts.

The decision tree terminates upon finding a factor.

Pre-factorisation, trim trailing zeros (in binary) or equiv is to divide by two until odd.

This is 1 and c for primes.

This is the square root for squares.

This is the pair of factors that are not 1 and c for the product of two primes that are not the same numbers (not a square).

This is the pair of factors closest to the square root for products of more than two factors (not including 1 and c as factors).

The first part.

Determine if a square.

Find the square root d and remainder e.

First decision, if e is 0, return d.

Second decision, if(GCD(e,d)!=1) return GCD(e,d);

That is the end of part one.

Part two.

Part two is recursive (since the grid is effectively an overlay of the fractal nature of integers).

A recursive part should immediately make sense in hindsight or will come as less of a shock in due course, considering that the complexity claim was at most big oh of the natural log of the length of c in bits.

The inputs into part two are to factorise d and e.

The end result of part two is a tree, descending from c with branches d and e, then branches descending from d and e or each of their square roots and remainders.