Higher-order recursive schemes (HORS) are schematic representations of functional programs. They generate possibly infinite ranked labelled trees and, in that respect, are known to be equivalent to a restricted fragment of the λY-calculus consisting of ground-type terms whose free variables have types of the form o → ⋯ → o (with o being a special case). In this paper, we show that any λY-term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. More precisely, for any λY-term M , there exists a HORS generating a tree that faithfully represents M 's (η-long) Böhm tree. In particular, the HORS captures higher-order binding information contained in the Böhm tree. An analogous result holds for finitary PCF. As a consequence, we can reduce a variety of problems related to the λY -calculus or finitary PCF to problems concerning higher-order recursive schemes. For instance, Böhm tree equivalence can be reduced to the equivalence problem for HORS. Our results also enable SO model-checking of Böhm trees, despite the general undecidability of the problem.