Abstract
In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size <= kappa it not only does not collapse kappa(+) but also preserves the strength of kappa (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf.
Dobrinen and Friedman (2010) [3], Friedman and Halilovic (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).