Give a bijective proofthe number of oriented increasing binary trees on the vertex set $\lbrace 1,2,\cdots,n\rbrace$ is the Eulerian number $E_n$(the number of alternating permutations in $\mathfrak{S}_n$). alternating permutation is downup permutation.
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I searched for the meanings of the words using Google, and I suspect that this document answers your question: www.stat.wisc.edu/~callan/notes/donaghey_bij/donaghey_bij.pdf 

