by $\mathcal L_n$ the set of Lyndon words with length $n$.

The standard right factor $v$ of a Lyndon word $w$ is its smallest proper suffix for the lexicographic order. Any Lyndon word $w$ can be written $uv$, in which $u$ is a Lyndon word and $v$ is the standard right factor of $w$. We call $uv$ the standard factorization of the Lyndon word $w$. Lyndon words were introduced by Lyndon [@Lyndon], to build a base of the free Lie algebra over $\mathcal A$. The standard factorization plays a central part in the building algorithm of this base. For each Lyndon word $w$, we can build a binary tree in the following manner: say that $w$ is the root, and has two children, that are the factors $u$ and $v$ of the standard factorization of $w$. Since $u$ and $v$ are still Lyndon words, they can also be divided into two standard factors which are their children and so on (see figure \[olebodessin\]). Then the average height of these trees characterizes the complexity of the building algorithm (see Chen, Fox and Lyndon [@Chen] and Lothaire [@Lothaire]). Thus the informations on the length of the standard right factor of a random 