Model checking trees generated by order-k Higher-Order Recursion Schemes (HORS) against Alternating Parity Tree-Automata (APT) is known to be a k-EXPTIME-complete problem (Ong’06). We exhibit a natural fragment of HORS, called tail-recursive HORS, and a restricted APT model, called bounded-alternation APT,such that the problem of model checking trees generated by order-k tail-recursive HORS against bounded-alternation APT is k-1-EXPSPACE-complete. The upper bound is achieved by converting the problem into an alternating reachability game, the lower one via a reduction from a tiling problem.