On Han's Hook Length Formulas for Trees

William Y.C. Chen, Oliver X.Q. Gao, Peter L. Guo

Abstract


Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to $k$-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give combinatorial proofs of Yang's formula for $k$-ary trees and the other formula of Han for binary trees. Our bijections are based on the structure of $k$-ary trees associated with staircase labelings.


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