Choose a factor pair of 111,111. Since the sum of the digits is equal to 6, which is divisible by 3, the number itself is divisible by 3. Also, since the number is a sequence of two 111s, the number is also divisible by 111. So, you can begin with the factor pair of 111 x 1001.
111,111
111 x 1001
3x37 7x143
_ __ __ 11 x 13
So, the prime factorization is 3 x 37 x 7 x 11 x 13, which is 3 x 7 x 11 x 13 x 37 when reordered in numerical order.
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11111137037,35291,7,3481,11,7,337,13,11,7,33 x 7 x 11 x 13 x 37 = 111111
Any multiple of 111111.
83 is prime. No tree.
61 is already prime. It doesn't need a tree.
When the bottom branch consists entirely of prime numbers.