  We study versions of the tree pigeonhole principle, $\mathsf{TT}^1$, in the
context of Weihrauch-style computable analysis. The principle has previously
been the subject of extensive research in reverse mathematics. Two outstanding
questions from the latter investigation are whether $\mathsf{TT}^1$ is
$\Pi^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf{RT}^1$,
and whether it is equivalent to any first-order statement of second-order
arithmetic. Using the recently introduced notion of the first-order part of an
instance-solution problem, we formulate, and answer in the affirmative, the
analogue of the first question for Weihrauch reducibility. We then use this, in
combination with other results, to answer in the negative the analogue of the
second question. Our proofs develop a new combinatorial machinery for
constructing and understanding solutions to instances of $\mathsf{TT}^1$.
