We introduce the polynomial-time tree reducibility
(ptt-reducibility). Our main result states that for
languages B and C it holds that
B ptt-reduces to C if and only if
the unbalanced leaf-language class of B is robustly contained in
the unbalanced leaf-language class of C.
This is the {\em unbalanced} analogue
of the well-known result by Bovet, Crescenzi, Silvestri, and Vereshchagin
which connects polylog-time reducibility with {\em balanced} leaf-languages.

We show that restricted to regular languages,
the levels 0, 1/2, 1, and 3/2 of the dot-depth hierarchy (DDH)
are closed under ptt-reducibility.
This gives evidence that with respect to unbalanced leaf-languages,
the dot-depth hierarchy and the polynomial-time hierarchy
perfectly correspond.
Level 0 of the DDH is closed under ptt-reducibility
even without the restriction to regular languages.
We show that this does not hold for higher levels.

As a consequence of our study of ptt-reducibility,
we obtain the first gap theorem of leaf-language definability
above the Boolean closure of \cNP:
If {\cal D} = \ULeaf({\cal C}) for some {\cal C} \subseteq \REG,
then {\cal D} \subseteq \bc(\cNP) or
there exists an oracle O such that
{\cal D}^O \not\subseteq \cP^{\cNP[\epsilon \cdot \log n]^O}
for every \epsilon < 1.