  The trie-based radix sort algorithm stores pairwise different infinite binary
strings in the leaves of a binary tree in a way that the Ulam-Harris coding of
each leaf equals a prefix (that is, an initial segment) of the corresponding
string, with the prefixes being of minimal length so that they are pairwise
different. We investigate the {\em radix sort tree chains} -- the tree-valued
Markov chains that arise when successively storing infinite binary strings
$Z_1,\ldots, Z_n$, $n=1,2,\ldots$ according to the trie-based radix sort
algorithm, where the source strings $Z_1, Z_2,\ldots$ are independent and
identically distributed. We establish a bijective correspondence between the
full Doob--Martin boundary of the radix sort tree chain with a {\em symmetric
Bernoulli source} (that is, each $Z_k$ is a fair coin-tossing sequence) and the
family of radix sort tree chains for which the common distribution of the $Z_k$
is a diffuse probability measure on $\{0,1\}^\infty$. In essence, our result
characterizes all the ways that it is possible to condition such a chain of
radix sort trees consistently on its behavior "in the large".
