Comment by wizzwizz4

This isn't true. Take 9_A = 1001_2. 28_A = 11100_2, which is 5 bits long (3 set). The biggest number in this path is 52_A = 110100_2, which is 6 bits long (3 set). 5 ≯ 6, and 3 ≯ 3: neither of my interpretations of your statement holds.