On Ward Numbers and Increasing Schr\"oder Trees

The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schr\"oder trees} by verifying they satisfy Ward's recurrence. We construct a direct type-preserving bijection between total partition trees and increasing Schr\"oder trees, complementing known type-preserving bijections to set partitions (including Chen's decomposition for increasing Schr\"oder trees). Weighted generalizations extend these bijections to enriched increasing Schr\"oder trees trees and Schr\"oder trees trees, yielding new links to labeled rooted trees. Finally, we deduce a functional equation for weighted increasing Schr\"oder trees, whose solution using Chen's decomposition leads to a combinatorial interpretation of a Lagrange inversion variant.