  The Horton-Strahler number of a rooted tree $T$ is the height of the tallest
complete binary tree that can be homeomorphically embedded in $T$. The number
of full binary trees with $n$ internal vertices and Horton-Strahler number $s$
is known to be the same as the number of Dyck paths of length $2n$ whose height
$h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$.
  In this paper, we present a new bijective proof of the above result, that in
fact strengthens and refines it as follows. We introduce a sequence of trees
$(\tau_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense
that $\tau_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge
0$, and $\tau_{i+1}$ strictly contains $\tau_i$ for all $i \ge 0$. Defining
$\mathcal{S}(T)$ to be the largest $i$ for which $\tau_i$ can be
homeomorphically embedded in $T$, we then show that the number of full binary
trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as
the number of Dyck paths of length $2n$ with height $h$. (We call
$\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.)
  Our proof is bijective and relies on a recursive decomposition of binary
trees (resp. Dyck paths) into subtrees with strictly smaller refined
Horton-Strahler number (resp. subpaths with strictly smaller height). In a
subsequent paper, we will show that the bijection has a continuum analogue,
which transforms a Brownian continuum random tree into a Brownian excursion and
under which (a continuous analogue of) the refined Horton-Strahler number of
the tree becomes the height of the excursion.
