First-difference order on a binary tree

A branch maximal under inclusion must end on level $3$, so it is determined by one of the eight words $000,001,010,011,100,101,110,111$. Conversely each such word is terminal and hence maximal. For distinct terminal words $s,t$, the comparison rule is $s<t⟺s(k)<t(k)\text{ for }k=\min \{i<3:s(i)\mathrm{\ne }t(i)\}\mathrm{.}$ The displayed list in the statement follows by sorting first by coordinate $0$, then by coordinate $1$, then by coordinate $2$.

The first-difference coordinates of consecutive branches are explicit:

The adjacent pair $000<001$ has no branch strictly between it, so this order is not dense. The elements $000$ and $111$ are respectively a first and last element, so it is not endpoint-free. Moreover $(000,010)=\{001\}$ is a nonempty open interval with the countable dense subset $\{001\}$; hence the nowhere-separable conclusion fails. Since the whole order has only eight elements, every antichain of pairwise disjoint nonempty open intervals is finite, so its ccc assertion holds only trivially.

The failed order-density and endpoint conclusions in step 2.1 come from the local successor order $\{0<1\}$: it has neither an intermediate successor nor successors on both sides. In [F1], replacing each successor set by a dense order without endpoints supplies exactly those insertions. By contrast, the failure of nowhere separability reflects bounded height: the argument in [F2] first bounds an arbitrary countable candidate dense set by a height $\delta$ and then splits above $\delta$, which is impossible in $2^{\le 3}$. Thus local dense successor orders yield density and no endpoints, while unbounded normal splitting is essential to the later nowhere-separability construction.

The root is the empty word, but no empty terminal branch occurs; at height zero it has exactly the two successors $0,1$. Every top word has length exactly three, so there is no hidden extension beyond the listed eight. All lists, comparisons, and witnesses are explicit finite data in ZF, and no choice principle is used.