Residues modulo powers of two in the Young-Fibonacci lattice

We study the subgraph of the Young-Fibonacci graph induced by elements with odd $f$-statistic (the $f$-statistic of an element $w$ of a differential graded poset is the number of saturated chains from the minimal element of the poset to $w$). We show that this subgraph is a binary tree. Moreover, the odd residues of the $f$-statistics in a row of this tree equidistibute modulo any power two. This is equivalent to a purely number theoretic result about the equidistribution of residues modulo powers of two among the products of distinct odd numbers less than a fixed number.