A Strahler bijection between Dyck paths and planar trees

The Strahler number of binary trees has been introduced by hydrogeologists and rediscovered in computer science in relation with some optimization problems. Explicit expressions have been given for the Strahler distribution, i.e. binary trees enumerated by number of vertices and Strahler number. Two other Strahler distributions have been discovered with the logarithmic height of Dyck paths and the pruning number of forests of planar trees in relation with molecular biology. Each of these three classes are enumerated by the Catalan numbers, but only two bijections preserving the Strahler parameters have been explicited: by Frann con between binary trees and Dyck paths, by Zeilberger between binary trees and forests of planar trees. We present here the missing bijection between forests of planar trees and Dyck paths sending the pruning number onto the logarithmic height. A new functional equation for the Strahler generating function is deduced. Some orthogonal polynomials appear, they are one parameter Tchebychee polynomials. 1 Strahler number of a binary tree We use the following classical notations for binary trees. A binary tree is a triple B = (L; r; R) or is reduced to an external vertex denoted by \". Here L and R are binary trees (left and right subtree respectively) and r denotes the root of B. In Figure 1, internal vertices are denoted by \". The number of binary trees with n internal vertices (and (n + 1) external vertices) is the Catalan number C n = 1 (n + 1) 2n n : Deenition 1. The Strahler number St(B) of the binary tree B is deened inductively by the relation