An example of generalized Schur operators involving planar binary trees
  Young's lattice is a prototypical example of differential posets. Differential posets have the Robinson correspondence, the correspondence between permutations and pairs of standard tableaux with the same shape, as in the case of Young's lattice. Fomin introduced generalized Schur operators to generalize the method of Robinson correspondence in differential posets to the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard tableaux with the same shape. In this paper, we introduce operators on the vector space whose basis is the set of planar binary trees. To prove that the operators are generalized Schur operators, we construct a correspondence, which is an extension of Fomin's r-correspondence for them. 
Yasuhide Numata