A classic proof of a recurrence for a very classical sequence (Q1374205)

The Schröder numbers \(s(n)\) satisfy the recurrence \(3(2n-1)s(n)= (n+1)s(n+1)+ (n-2)s(n- 1)\), \(n\geq 2\), with \(s(1)= s(2)=1\). The authors give a direct combinatorial proof of the recurrence by establishing a bijection from the set \(\{1,2,3\}\times\text{PT}(n)\) onto the disjoint union \(\text{LT}(n+ 1)\dot\cup\text{IT}(n- 1)\) where \(\text{PT}(n)\), \(\text{LT}(n)\), \(\text{IT}(n)\) are the sets of pointed, leaf-pointed and interior pointed well-weighted trees with \(n\) leaves respectively.