  We investigate the tree-to-tree functions computed by
\enquote{affine$\lambda$-transducers}: tree automata whose memory consists of
an affine $\lambda$-term instead of a finite state. They can be seen as
variations on Gallot, Lemay and Salvati's Linear High-Order Deterministic Tree
Transducers. When the memory is almost purely affine (\textit{\`a la}
Kanazawa), we show that these machines can be translated to tree-walking
transducers (and with a purely affine memory, we get a reversible tree-walking
transducer). This leads to a proof of an inexpressivity conjecture of
\titocecilia on \enquote{implicit automata} in an affine $\lambda$-calculus.
The key technical tool in our proofs is the Interaction Abstract Machine (IAM),
an operational avatar of the \enquote{geometry of interaction} semantics of
linear logic. We work with ad-hoc specializations to (almost) affine
$\lambda$-terms of a tree-generating version of the IAM.
